SPK Required Reading

This required reading document is reproduced from the original NAIF document available at https://naif.jpl.nasa.gov/pub/naif/misc/toolkit_docs_N0067/C/req/spk.html

Note

These required readings documents were translated from documentation for N67 CSPICE. These pages may not be updated as frequently as the CSPICE version, and so may be out of date. Please consult the changelog for more information.

Important

NOTE any functions postfixed by "_" mentioned below are Fortan-SPICE functions unavailable in SpiceyPy as the NAIF does not officially support these with "_c" function wrappers within the CSPICE API. If these functions are necessary for your work please contact the NAIF to request that they be added to the CSPICE API

Abstract

The SPK system is the component of SPICE concerned with ephemeris data.

Purpose

The purpose of this document is to describe the SPICE Toolkit software provided in the software library CSPICE, (C SPICE library) used for producing and accessing SPICE ephemeris data. In addition this document describes SPK---the common file format for NAIF's S-kernel and ephemeris portion of the P-kernel.

Intended Audience

This document is intended for all users of SPK (ephemeris) kernel files.

References

All references are to NAIF documents. The notation [Dn] refers to NAIF document number.

1. [349] Frames Required Reading (frames)

2. [174] CK Required Reading (ck)

3. [254] PCK Required Reading (pck)

4. [222] Spacecraft Clock Time Required Reading (sclk)

5. [218] KERNEL Required Reading (kernel)

6. [219] NAIF IDS Required Reading (naif_ids)

7. [163] JPL Internal Memorandum on Modified Difference Array polynomials; F. Krogh

8. [164] Precession Matrix Based on IAU (1976) System of Astronomical Constants; E. M. Standish; Astronomy and Astrophysics 73, 282-284 (1979)

9. [165] Orientation of the JPL Ephemerides, DE200/LE200, to the Dynamical Equinox of J2000; E. M. Standish; Astronomy and Astrophysics 114, 297-302 (1982)

10. [166] The JPL Asteroid and Comet Database (as Implemented by NAIF); a collection of papers and memos; assembled by I. Underwood; 11 Dec 1989

11. [167] Double Precision Array Files (DAF) - Required Reading; latest version (daf.req)

12. [212] COMMNT User's Guide (commnt.ug)

DAF Run-Time Binary File Format Translation

Starting with the N0052 release of the SPICE Toolkit (January, 1) certain supported platforms are able to read DAF-based binary files (SPK, CK and binary PCK) that were written using a different, or non-native, binary representation. This access is read-only; any operations requiring writing to the file (adding information to the comment area, or appending additional ephemeris data, for example) require prior conversion of the file to the native binary file format. See the Convert User's Guide, convert.ug, for details.

Detection of Non-native Text Files

Starting with the N0057 release of the SPICE Toolkit (March, 2004) the SPICE data loading mechanism detects and prohibits loading text kernel files containing lines terminated with EOF character(s) non-native to the platform on which the Toolkit was compiled. If a non-native EOL terminator is detected in the first 132 characters of a text kernel, the execution is stopped and an error message is displayed. This feature does not work with files that are smaller than 132 bytes or have the first line longer than 132 characters.

If you're in a hurry

We'll discuss things in more detail in a moment but in case you are just looking for the right name of the function to perform some ephemeris task, here is a categorization of the most frequently used SPK and related functions in CSPICE. Input arguments are given in lower case and enclosed in angle brackets. Output arguments are given in plain lower case.

High Level Functions

Loading/Unloading an SPK file

• furnsh()

• unload()

Getting coverage summary

• spkobj()

• spkcov()

Retrieving states (position and velocity) using names of objects

• spkezr()

Retrieving positions using names of objects

• spkpos()

Retrieving states using NAIF ID codes

• spkez()

• spkgeo()

Retrieving positions using NAIF ID codes

• spkezp()

• spkgps()

Calculating Uplink and Downlink Light Time

• ltime()

Loading/Unloading Binary PCK files (see PCK Required Reading, pck)

• furnsh()

• unload()

Loading Text based kernels---PCK, SCLK, etc.

• furnsh()

Loading/Unloading C-kernels (see CK Required Reading, ck)

• furnsh()

• unload()

Foundation Functions

The functions listed in this section are the real work horses of the SPK and related systems. Not all of the functions in this section are described in this document. In those cases, the appropriate SPICE document is cited.

Selecting files and segments

• spksfs()

Computing states from segment descriptors

• spkpvn()

Correcting for stellar aberration

• stelab()

Translating between object names and object ID codes (see NAIF_IDS Required Reading, naif_ids)

• bodn2c()

• bodc2n()

Translating between frame names and frame ID codes (see Frames Required Reading, frames)

• frmnam()

• namfrm()

State transformation matrices (see Frames Required Reading, frames)

• sxform()

Classifying frames (see Frames Required Reading, frames)

• frinfo()

Utility Programs

Examining SPK files

brief
commnt
spacit

Converting to and from transfer format

spacit
tobin
toxfr

Introduction

To help fully understand the science data returned from a spacecraft's instruments it is necessary to know, at any given epoch, the positions and possibly the velocities of the spacecraft and all the target bodies of interest. The purpose of the SPK---which stands for S(pacecraft) and P(lanet) Kernel---file is to allow ephemerides for any collection of solar system bodies to be combined under a common file format, and accessed by a common set of functions.

Historically, ephemerides for spacecraft have been organized differently from those for planets and satellites. They are usually generated through different processes and using different representations. However, there is no essential reason for keeping them separate. A spacecraft, planet, satellite, comet, or asteroid has a position and velocity relative to some center of mass and reference frame. Consequently all of these objects can be represented in an SPK file.

Consider the Galileo mission. Some of the objects of special interest to the Galileo mission are:

Galileo Spacecraft
Galileo Probe
Earth
Moon
Earth Moon Barycenter
Venus
Sun
Solar System Barycenter (S.S.B.)
Asteroid Ida
Ida's Satellite Dactyl
Asteroid Gaspra
Comet Shoemaker-Levy
Jupiter System Barycenter (J.B.)
Jupiter
Io
Ganymede
Europa
Callysto
Goldstone Tracking Station.

Each of these objects has a position and velocity (state) relative to some other object. The graph below illustrates which objects will be used as reference objects for representing the states of others.

                       +Gll
                      /             probe
                     /               |    o Comet
             Gaspra /             Gll+   /  Shoemaker Levy
      Gll +--o     /                  \ /
             |    /   Venus    Jupiter o--probe
             |   /      o--+           |
 Gll +       |  /      /   Gll         |  Io
     |       | /      /                |  o-----+Gll
     |       |/      /             J.B.| /
Ida  o-------o------o------------------o ----o------+Gll
    /         Sun   S.S.B.            / \    Europa
   o                 \      Ganymede /   \
Dactyl                \             o     \
                       \            |      o Callisto
 Earth-Moon Barycenter  o----o      +      |
                        |   Moon    Gll    |
                        |                  + Gll
                        o Earth
                       / \
                      /   \
                     /     + Gll
                    o
                 Goldstone

This graph is somewhat complicated. Nevertheless, the complete ephemeris history for all of these objects can be captured in a single SPK file. (Although we can store the entire ephemeris history illustrated above in a single SPK file, for the sake of data management a project is likely to use several SPK files. However, even in this case, all of the SPK files can be used simultaneously.)

The SPK format is supported by a collection of functions that are part of the CSPICE library---the major component of the SPICE Toolkit. This family of SPK functions provides the following capabilities:

1. Insert ephemeris data from some source into an SPK file.

2. Make the ephemeris data in one or more SPK files available to a user's program.

3. Return the apparent, true, or geometric state (position and velocity) of one ephemeris object as seen from another in some convenient reference frame.

The SPK software allows you to ignore the potential ephemeris complexity associated with the a mission such as Galileo and allows you to more directly compute various quantities that depend upon the position or velocity of one object as seen from another.

SPK Files

SPICE software writes SPK files in a binary (non-ASCII) format structured in a NAIF developed abstract file architecture called Double Precision Array File (DAF). The DAF architecture and supporting software is discussed in the DAF Required Reading document, daf.req. The SPICE file identification word occupying the first eight bytes of a properly created binary SPK file is DAF/SPK . For more information on SPICE identification words refer to the Kernel Required Reading document, kernel. If you need only use SPK files as a data source or if you will use a SPICE utility program for creating SPK files, you can safely ignore aspects of the DAF system not covered by this document. On the other hand, if you plan to write software for creating SPK files you will probably need to familiarize yourself with the DAF software contained in CSPICE. The particular aspects of the DAF architecture that are relevant to the SPK format are discussed later in this document (see below---SPK Format).

Since SPKs are written as binary files, the specific binary format depends on the computer architecture on which the SPK was created, in the case of SPICE either big-endian or little-endian (NAIF no longer supports DEC platforms).

Use of SPK files between computers

NAIF extended the DAF capability in SPICE Toolkit delivery N0052 to allow reading of both big-endian and little-endian binary DAF files by all toolkits. This process is a run-time interpretation of non-native binary files. Run-time interpretation does not yet work for any file built upon the SPICE "DAS" architecture.

NAIF provides two utility programs---TOXFR and SPACIT for converting SPICE binary kernels to a transfer format suitable for text copying from one computer to another. Once the transfer format file has been copied, the SPICE utilities TOBIN and SPACIT are available for converting the transfer format file to the binary format suitable for the new machine.

The utilities TOXFR and TOBIN are command line programs. To convert a binary kernel to transfer format you simply type TOXFR followed by the name of the binary kernel at your terminal prompt.

prompt> toxfr spk_file

To convert a transfer format to binary format, you type TOBIN followed by the name of the transfer format kernel. .. code-block:: text

prompt> tobin transfer_file

The utility SPACIT is an interactive program that allows you to select a function from a menu to perform on a file. This program can also be used to convert to or from transfer format files. Note that transfer format files cannot be loaded into a SPICE based program to retrieve ephemeris data. Only binary format files can be used for retrieving ephemeris data with SPICE software.

CSPICE (and by extension Icy and Mice) uses the same binary kernels as does SPICELIB.

Examining SPK files

Since SPK files are binary files, you can't just open them with your favorite text editor to determine which ephemeris objects are represented in the file. Instead you need to use one of the SPICE utility programs that allow you to summarize the ephemeris contents of an SPK file. The first of these is SPACIT which was introduced above. The second is the command line utility BRIEF.

BRIEF gives a quick summary of the contents of the file and supports a wide set of summary options. SPACIT on the other hand, provides summaries that are more detailed and reflect closely the actual internal structure of the file. Unless you need the more detailed summary, you'll probably find BRIEF to be a better tool for examining the contents of an SPK file.

Meta Data in the SPK file

SPICE kernels may contain meta data that describe the contents, intended use, accuracy, etc. of the kernel. This meta data is called the comments portion of the kernel. Many SPK files contain comments that can help you decide upon the suitability of the kernel for your application. Two SPICE utilities are available for examining the comments of a binary kernel---COMMNT and SPACIT.

We've already introduced SPACIT. COMMNT is similar to SPACIT in that it too is an interactive program. However, COMMNT also allows you to modify the comments of an SPK file. Using COMMNT you can delete the comments of an SPK file, extract the comments to a text file, or append the text from some text file to the comments already present in the kernel.

If you create SPK files, we strongly recommend that you add comments to the kernel that describe who created it, expected usage of the kernel, and the expected accuracy of the position/velocity information contained in the kernel. A comment template is provided in the appendix COMMENTS.

Warning: If you add comments to an SPK (or other binary kernel) using COMMNT, you must wait for the program to complete the task before exiting the program. Failure to wait for COMMNT to finish its work will result in irreparable corruption of the binary kernel. (See the COMMNT User's Guide, commnt.ug, [212] for details on the use of COMMNT).

Terminology

Throughout this document we shall be using terms such as reference frame, state, ephemeris time, etc. We include a brief review of these terms below.

Reference Frame

A reference frame is a Cartesian coordinate system with three axes---x, y and z. The axes are mutually orthogonal. The center of the frame is the origin of the Cartesian reference system. For the reference frames in SPICE, the positions of the axes are typically defined by some observable object. For example, in the J2000 reference frame, the x-axis is defined to lie in the intersection of two planes: the plane of the Earth's equator and the plane of the Earth's orbit. The z-axis is perpendicular to the Earth's equator. The y-axis completes a right-handed system. The center of the frame is typically taken to be the solar system barycenter. (Note we are not attempting to rigorously define the J2000 frame here. We are only illustrating how reference frames are defined. Many more details are required for a rigorous definition of the J2000 frame. These details are given in the SPICE document Frames [349].)

State

A state is an array of six double precision numbers. The first three numbers give the x, y, and z coordinates respectively for the position of some object relative to another object in some Cartesian reference frame. The next three numbers give the velocity ( dx/dt, dy/dt and dz/dt respectively) of the object with respect to the same reference frame.

Inertial Frame

An inertial frame, is one in which Newton's laws of motion apply. A frame whose axes are not moving with respect to the observed positions of distant galaxies and quasars approximates an inertial frame.

Non-Inertial Frame

A non-inertial frame is a frame that rotates with respect to the celestial background. For example a frame whose axes are fixed with respect to the features on the surface of the Earth is a non-inertial frame.

Ephemeris Time (ET)

Ephemeris time, ET, is the independent variable in the equations of motion that describe the positions and velocities of objects in the solar system. In CSPICE we treat ET as a synonym for Barycentric Dynamical Time. As far as has been experimentally determined, an atomic clock placed at the solar system barycenter, would provide a faithful measure of ET.

Seconds Past 2000

In the SPK system times are specified as a count of seconds past a particular epoch---the epoch of the J2000 reference frame. This reference epoch is within a second or two of the UTC epoch: 12:01:02.184 Jan 1, 2000 UTC. (See the document time.req for a more thorough discussion of the J2000 epoch). Epochs prior to this epoch are represented as negative numbers. The units of ET are designated in several different ways: seconds past 2000, seconds past J2000, seconds past the Julian year 2000, seconds past the epoch of the J2000 frame. All of these phrases mean the same thing and are used interchangeably throughout this document.

SPK segment

The trajectories of objects in SPK files are represented in pieces called segments. A segment represents some arc of the full trajectory of an object. Each segment contains information that specifies the trajectory of a particular object relative to a particular center of motion in a fixed reference frame over some particular interval of time. From the point of view of the SPK system segments are the atomic portions of a trajectory.

The SPK Family of Functions

CSPICE contains a family of functions that are designed specifically for use with SPK files. The name of each function begins with the letters spk, followed by a two- or three-character mnemonic. For example, the function that returns the state of one body with respect to another is named spkezr(), pronounced S-P-K-easier. A complete list of mnemonics, translations, and calling sequences can be found at the end of this document.

Each function is prefaced by a complete CSPICE header, which describes inputs, outputs, restrictions, and exceptions, discusses the context in which the function can be used, and shows typical examples of its use. Any discussion of the functions in this document is intended as an introduction: the final documentation for any function is its header.

Whenever an SPK function appears in an example, the translation of the mnemonic part of its name will appear to the right of the reference, in braces. We also continue with the convention of distinguishing between input and output arguments by enclosing input arguments in angle brackets. For example,

state, lt = spkezr( <targ>,  <et>,  <frame>, <aberr>, <obs>)  # Easier state

All C functions, including those whose names do not begin with SPK, are from CSPICE or the standard ANSI C library. SPK readers are available to perform the following functions.

1. Determine the apparent, true, or geometric state of a body with respect to another body relative to a user specified reference frame.

2. Determine the apparent, true, or geometric state of a body with respect to an observer having a user-supplied state.

3. Determine the geometric state of a body with respect to the solar system barycenter.

4. Determine the geometric state of a target body with respect to its center of motion for a particular segment.

5. Determine, from a list of SPK files supplied by the calling program, the files and segments needed to fulfill a request for the state of a particular body.

Computing States

spkezr() is the most powerful of the SPK readers. It determines the apparent, true, or geometric state of one body (the target) as seen by a second body (the observer) relative to a user specified reference frame.

state, lt = spkezr( <targ>,  <et>,  <frame>, <aberr>, <obs>)  #  Easier state

The function accepts five inputs---target body, epoch, reference frame, aberration correction type, and observing body---and returns two outputs---state of the target body as seen from the observing body, and one-way light-time from the target body to the observing body. The target body, observing body and frame are identified by strings that contain the names of these items. For example, to determine the state of Triton as seen from the Voyager-2 spacecraft relative to the J2000 reference frame

state, lt = spkezr( "triton",  et,  "j2000", aberr, "voyager-2" ) #  Easier state

By definition, the ephemerides in SPK files are continuous: the user can obtain states at any epoch within the interval of coverage. Epochs are always specified in ephemeris seconds past the epoch of the J2000 reference system (Julian Ephemeris Date 2451545.0 ) For example, to determine the state of Triton as seen from Voyager-2 at Julian Ephemeris Date 2447751.8293,

et = ( 2447751.8293 - j2000_c() ) * spd_c()

state, lt = spkezr( "triton", et, "j2000", <aberr>,  "voyager-2" ) #  Easier state

where the function j2000() returns the epoch of the J2000 frame (Julian Ephemeris Date 2451545.0) and the function spd() returns the number of seconds per Julian day (86400.0). The ephemeris data in an SPK file may be referenced to a number of different reference frames. States returned by spkezr() do not have to be referenced to any of these native frames. The user can specify that states are to be returned in any of the frames recognized by the frame subsystem. For example, to determine the state of Triton as seen from Voyager-2, referenced to the J2000 ecliptic reference frame,

state, lt = spkezr( "triton", et, "eclipj2000", aberr, "voyager-2") #  Easier state

spkezr() returns apparent, true, or geometric states depending on the value of the aberration correction type flag aberr. Apparent states are corrected for planetary aberration, which is the composite of the apparent angular displacement produced by motion of the observer (stellar aberration) and the actual motion of the target body (correction for light-time). True states are corrected for light-time only. Geometric states are uncorrected.

Instead of using the potentially confusing terms true and geometric to specify the type of state to be returned, spkezr() requires the specific corrections to be named. To compute apparent states, specify correction for both light-time and stellar aberration: LT+S. To compute true states, specify correction for light-time only: LT. To compute geometric states, specify no correction: NONE.

In all cases, the one-way light-time from the target to the observer is returned along with the state.

Computing States using Constant-Velocity or Constant-Position Objects

Objects such as tracking stations, rover or spacecraft components, or fixed surface points can be treated by the SPK subsystem as ephemeris objects just as easily as bodies such as planets and natural satellites. For example, using an SPK file for the geocentric location of a tracking station enables spkezr() to compute states of targets relative to the tracking station, providing all needed kernel data have been loaded.

However, it is not always convenient to create an SPK file to provide data for an ephemeris object, particularly when that object's location is known only at run time.

For an object that has constant velocity, relative to a specified center of motion, in a specified reference frame, CSPICE offers a set of functions to compute states relative to other ephemeris objects, where the other objects have ephemeris data provided by SPK files:

spkcpo_c  # SPK, constant position observer state
spkcpt_c  # SPK, constant position target state
spkcvo_c  # SPK, constant velocity observer state
spkcvt_c  # SPK, constant velocity target state

The constant position routines have simplified interfaces; these handle the special case where the constant velocity is zero. Each of the above functions requires that sufficient SPK data be available to compute the state of the center of motion, relative to the other ephemeris object, of the constant-velocity or constant-position object.

States computed by SPK functions for constant-velocity or constant-position objects optionally can be corrected for light time and stellar aberration, just as is done by spkezr().

A limitation of representing objects using constant velocities or positions, instead of creating SPK files to provide the ephemerides of those objects, is that high-level CSPICE geometry routines such as sincpt() or subpt() cannot work with such objects---these functions require SPK data for all ephemeris objects participating in the computations they perform.

The Computation of Light Time

The light time corrected position component of a state vector returned by the SPK system is the 3-vector difference

TARGET_SSB ( ET + S*LT )  - OBSERVER_SSB ( ET )

where TARGET_SSB and OBSERVER_SSB give the position of the target and observer relative to the solar system barycenter, and where S is -1 for reception corrections (where light travels from the target to the observer) and 1 for transmission corrections (where light travels from the observer to the target). LT is the unique number that satisfies:

      | TARGET_SSB ( ET + S*LT )  -  OBSERVER_SSB ( ET ) |
LT =  ----------------------------------------------------
                        Speed of Light

where .. code-block:: text

position |

indicates the length of a position vector. The velocity portion of the returned state is the derivative with respect to the observation time ET of the light time corrected position.

Mathematically, LT can be computed to arbitrary precision via the following algorithm:

LT_0 = 0



        | TARGET_SSB ( ET - LT_(i-1) ) - OBSERVER_SSB ( ET ) |
LT_i =  ------------------------------------------------------
                           Speed of Light


   for i = 1, ...

It can be shown that the sequence LT_0, LT_1, LT_2, ... converges to LT geometrically. Moreover, it can be shown that the difference between LT_i and LT satisfies the following inequality.

                              i
| LT - LT_i | < LT_i * ( V/C )  / ( 1 - V/C )
   for i = 1, ...

where V is the maximum speed of the target body with respect to the solar system barycenter and C is the speed of light.

Precision of Light Time Computations

Let's examine the error we make if we use LT_2 as an approximation for LT. This is an analysis of precision; we'll ignore errors in the data and those in the input times.

For nearly all objects in the solar system V is less than 60 km/sec. The value of C is approximately 300000 km/sec. Thus V/C is 2.0E-4, and the one iteration solution for LT (in which the target-SSB vector is corrected once) has a potential relative error of not more than 4.0E-8. This is a potential light time error of approximately 2.0E-5 seconds per astronomical unit of distance separating the observer and target. Thus as long as the observer and target are separated by less than 50 Astronomical Units, the error in the light time returned using option LT is less than 1 millisecond.

For this reason, CSPICE uses LT_2 to approximate LT when you request a light time corrected state by setting the aberration correction argument in spkezr() to any of LT, XLT, LT+S, XLT+S.

The maximum error in the light time corrected target-SSB position vector is larger by a factor of C/V than V times the maximum relative light time error. This is because the (i-1)st light time estimate is used to compute the ith estimate of target-SSB position vector. Given the assumptions above, the maximum position error for the LT-style correction is bounded by

60 km/s * (1/(2.0E-4)) * 2*1.0E-5 s / AU

or 6 km per astronomical unit of distance separating the observer and target. In practice, the difference between positions obtained using one-iteration and converged light time is usually much smaller than the value computed above and can be insignificant. For example, for the spacecraft Mars Reconnaissance Orbiter and Mars Express, the position error for the one-iteration light time correction, applied to the spacecraft-to-Mars center vector, is approximately 2 cm.

You can make spkezr() (and other applicable SPK functions) compute a better approximation to LT and compute more accurate light-time corrected states by commanding that it compute a converged Newtonian value for LT. To do this set the light time portion of the aberration correction specification to CN (the possible such aberration correction specifications are`CN', XCN, CN+S, or XCN+S). spkezr() will then return a converged value, usually equal to LT_4, as the approximation for light time; the returned state will be converged as well. Then the maximum error in LT_4 is less than

1.0E-3 * (V/C)**2 seconds

which is less than 4e-11 seconds for any observer/target pair in the solar system that satisfies the assumptions above. The corresponding position error bound is 1.2 cm at a separation of 50 AU. However, you should note that this is a purely Newtonian approximation to the light time. To model the actual light time between target and observer one must take into account effects due to General relativity. These may be as high as a few hundredths of a millisecond for some geometric cases.

The functions in the SPK family do not attempt to perform either general or special relativistic corrections in computing the various aberration corrections. For many applications relativistic corrections are not worth the expense of added computation cycles. If, however, your application requires these additional corrections we suggest you consult the astronomical almanac (page B36) for a discussion of how to carry out these corrections.

Light Time Corrected Non-Inertial States

When we observe a distant object, we don't see it as it is at the moment of observation. We see it as it was when the photons we have sensed were emitted by or reflected from the object. Thus when we look at Mars through a telescope, we see it not as it is now, but rather as it was one light-time ago. This is true not only for the position of Mars, but for its orientation as well.

Suppose that a large balloon has been launched into the Martian atmosphere and we want to determine the Mars bodyfixed state of the balloon as seen from Earth at the epoch ET. We need to determine both the light time corrected position of the balloon, and the light time corrected orientation of Mars. To do this we compute two light times. The light time to the center of the Mars bodyfixed frame (i.e. the center of Mars) and the light time to the balloon. Call the light time to the center of the Mars frame LT_F and call the light time to the balloon LT_T. The light time corrected state of the balloon relative to the Mars bodyfixed frame is the location of the balloon at ET - LT_T in the bodyfixed frame of Mars as oriented at ET - LT_F.

spkezr() carries out all of these computations automatically. In this case the computation would be computed by a function call similar to this:

spkezr( "mars_balloon",  <et>,  "iau_mars", "lt",
           "earth",         state, lt              )

spkezr() uses the following rules when computing states.

1. When no corrections are requested from spkezr() (ABCORR = 'NONE'), the state of the target is determined at the request time ET and is represented in the specified reference frame as it is oriented at time ET.

2. When light time corrections are requested from spkezr() (ABCORR = 'LT'), two light times are determined: LT_F the light time to the center of the specified reference frame, and LT_T the light time to the target. The state of the target is given as it was at ET - LT_T in the frame as it was oriented at ET - LT_F.

3. When light time and stellar aberrations are requested from spkezr() (ABCORR = 'LT+S'), both LT_F and LT_T are again computed. The state of the target at ET - LT_T is corrected for stellar aberration and represented in the reference frame as it was oriented at ET - LT_F.

4. Light-time corrected velocities are computed taking into account the rate of change of light time both between observer and target and between observer and the center of the non-inertial frame. The rate of change of the target frame's orientation is accounted for as well.

In the actual implementation of spkezr() a few short cuts are taken. When light time requested states relative to an inertial frame are requested, the orientation of the frame is not corrected for light time. The orientation of an inertial frame at ET - LT_F is the same as the orientation of the frame at ET. Computations involving inertial frames take advantage of this observation and avoid redundant computations.

An example

Here we illustrate how you could use spkezr() together with other CSPICE functions to determine if at a particular epoch ET the Mars Global Surveyor spacecraft is occulted by Mars.

We will need the lengths of the axes of the triaxial ellipsoid that is used to model the surface of Mars. Either of the CSPICE functions bodvcd() or bodvrd() will retrieve this information from a loaded PCK file. bodvrd() uses the name of the body, while bodvcd() uses the NAIF ID code for Mars (499) to retrieve the lengths of the axes. We may call bodvcd() as shown:

nvals, axes =bodvcd( 499, "RADII", 3 )

a = axes[0];
b = axes[1];
c = axes[2];

Next we compute the state of Mars relative to Earth and the state of MGS relative to Earth in the Mars bodyfixed frame.

marsst, lt = spkezr( "mars", et, "iau_mars", "lt+s",  "earth")
mgsst,  lt = spkezr( "mgs",  et, "iau_mars", "lt+s", "earth") # Easier State

Compute the apparent position of the Earth relative to Mars in the apparent Mars bodyfixed frame. This means simply negating the components of marsst. The CSPICE function vminus() carries out this task.

vminus( marsst, estate )

Determine if the line of sight from Earth to MGS intersects the surface of Mars. The CSPICE function surfpt() will find this intersection point if it exists.

surfpt( estate, mgsst, a, b, c, point, &found )

Finally, if a point of intersection was found, was it between the Earth and the MGS spacecraft. To find out we can compare the distances between the intersection point and the spacecraft. The CSPICE function vnorm() computes the length of the vector from Earth to MGS. The function vdist() computes the distance between the point and the Earth.

if ( found )
   {
   betwn = (  vdist ( estate, point ) < vnorm ( mgsst )  )
   }
else
   {
   betwn = SPICEFALSE;
   }


if ( betwn )
   {
   printf ( "mgs is behind mars\n" )
   }
else
   {
   printf ( "mgs is not behind mars\n" )
   }

Integer ID Codes Used in SPK

Low level SPK software uses integer codes to identify ephemeris objects, reference frames and data representation, etc. At low levels of the SPICE system only integer codes are used to communicate information about objects. To some extent, these codes are a historical artifact in the design of the SPICE system. Nevertheless, these integer codes provide economies in the development of SPICE software.

High-level SPICE software uses names (character strings) to refer to the various SPICE objects and translates between names and integer codes. Thus to some extent you can disregard the integer codes used by the SPICE internals. However, occasionally, due to the introduction of new ephemeris objects, the name translation software will be unable to find a name associated with an ID code. To retrieve states for such an object you will need to use the integer code for the object in question. If you are using spkezr(), you can supply this integer code as a quoted string. For example the following two function calls will both return the state of TRITON as seen from Voyager-2. (The NAIF integer code for TRITON is 801; the NAIF integer code for Voyager 2 is -32).

state, lt = spkezr( "triton", et, "eclipJ2000", aberr,    "voyager-2") #  Easier state
state, lt = spkezr( "801", et, "eclipJ2000", aberr,   "-32" ) #  Easier state

Consult the NAIF IDS Required Reading file, naif_ids, for the current list of body codes recognized by the SPICE Toolkit software.

spkez() ------------------------------------------------------------------------------------------------------------^^^^^^^^^^^^^^^^^^^^

spkezr() relies upon two lower level functions that may be useful under certain circumstances.

The function spkez() performs the same functions as spkezr(). The only difference is the means by which objects are specified. spkez() requires that the target and observing bodies be specified using the NAIF integer ID codes for those bodies.

state, lt = spkez( <targ_id>, <et>, <frame>, <corr>, <obj_id> ) #  SPK Easy

The NAIF-ID codes for ephemeris objects are listed in the NAIF_IDS required reading file, naif_ids. spkez() is useful in those situations when you have ID codes for objects stored as integers. There is also a modest efficiency gain when using integer ID codes instead of character strings to specify targets and observers.

The function spkgeo() returns only geometric (uncorrected) states. The following two function calls are equivalent.

state, lt = spkez( <targ_id>,  <et>,   <frame>, "none", <obj_id>  ) # SPK Easy

state, lt = spkgeo( <targ_id>,  <et>,   <frame>, <obj_id> ) # SPK Geometric

spkgeo() involves slightly less overhead than does spkez() and thus may be marginally faster than calling spkez().

Loading Files

Note that spkezr(), spkez() and spkgeo() do not require the name of an SPK file as input. These functions rely on the lower level routine in the SPK subsystem to maintain a database of ephemeris files. Your application program indicates which files are to be used by passing their names to function furnsh() -- generic loader that can be used to load SPICE kernel files of any type.

for e in ephem
    spice.furnsh(e)        # Load kernel file
# or more simply:
spice.furnsh(ephem)

In general, a state returned by spkezr() is built from several more primitive states. Consider the following diagram, which shows some of the states that might be needed to determine the state of the Galileo spacecraft as seen from Earth:

         Jupiter_Barycenter --- Europa
         /                       \
        /                         \
       /                          Spacecraft
      /
     /
    /
   /
SSB
   \
    \
     \
     EMB --- Earth

(SSB and EMB are the solar system and Earth-Moon barycenters.) Each state is computed from a distinct segment. The segments may reside in a single SPK file, or may be contained in as many as five separate files. For example, the segments needed to compute the Earth-spacecraft state shown above might come from the following set of files:

furnsh( "barycenters.bsp"    )  # Load kernel file
furnsh( "planet-centers.bsp" )  # Load kernel file
furnsh( "satellites.bsp"     )  # Load kernel file
furnsh( "spacecraft.bsp"     )  # Load kernel file

or from the following set:

furnsh( "earth.bsp"      )      # Load kernel file
furnsh( "jupiter.bsp"    )      # Load kernel file
furnsh( "spacecraft.bsp" )      # Load kernel file

Data Precedence

An SPK file may contain any number of segments. A single file may contain overlapping segments: segments containing data for the same body over a common interval. When this happens, the latest segment in a file supersedes any competing segments earlier in the file. Similarly, the latest file loaded supersedes any earlier files. In effect, several loaded files become equivalent to one large file.

Unloading Files

The number of SPK files that may be loaded at any one time is limited but very large -- up to 5000 total for all loaded SPK, CK, and binary PCK files combined. Although unlikely, in some cases your application program may need to unload some SPK files to make room for others or to remove a particular SPK from the set of loaded data. An SPK file may be unloaded by supplying its name to function unload() -- generic unloader that can be used to unload SPICE kernel of any type. The sequence of statements shown below,

furnsh( "file.a" )     # Load kernel file
furnsh( "file.b" )     # Load kernel file
furnsh( "file.c" )     # Load kernel file
unload( "file.b" )     # Unload kernel file
furnsh( "file.d" )     # Load kernel file
unload( "file.c" )     # Unload kernel file

is equivalent to the following (shorter) sequence:

furnsh( "file.a" )     # Load kernel file
furnsh( "file.d" )     # Load kernel file

Getting Coverage Summary

The CSPICE includes two functions for obtaining information about the contents of an SPK file from within an application.

The spkobj() function provides an API via which an application can find the set of bodies for which a specified SPK file contains data. The body IDs are returned in a SPICE set data structure (see sets.req).

The spkcov() function provides an API via which an application can find the time periods for which a specified SPK file provides data for an body of interest. The coverage information is a set of disjoint time intervals returned in a SPICE window data structure (see other stuff tutorial and windows.req).

Refer to the headers of spkobj() and spkcov() for details on the use of those routines.

Loading Auxiliary Files

Prior to the inclusion of non-inertial frames in the SPK system, the states of objects computed by the SPK system required only that you load the correct SPK files and call the correct functions. The inertial frame transformations needed for converting from one inertial frame to another are hard wired into the SPICE system. The transformations are part of the object code of the CSPICE library---no additional data need be supplied to compute these transformations. This approach to carrying out inertial frame transformations was chosen because the transformations are compactly represented and do not change as the result of further observations. They are essentially definitions.

On the other hand, the orientation of non-inertial frames with respect to other frames are almost always the result of observation. They are improved and extended as further observations are made. For some of these frames (such as spacecraft fixed frames) very large data sets are needed to express the orientation of the frame with respect to other frames. Frame transformations that are a function of time and require megabytes of data are not suitable for encapsulation in C or FORTRAN source code. As a result, in the SPICE system, the computation of non-inertial frame transformations depends upon data stored in other SPICE kernels. If you request states relative to a non-inertial frame or use ephemerides that are represented relative to non-inertial frames you must load additional SPICE kernels. The method by which an auxiliary kernel is loaded depends upon the type of the kernel.

There are currently five classes of reference frames that are supported by the SPICE system. We give a brief overview of these frames here. For a more thorough discussion of the various types of frames see the recommended reading file frames.

Inertial frames

• Inertial frames are built into the SPICE system. You don't need to do anything to make their definitions available to your program. Inertial frames have NAIF ID codes whose values are in the range from 1 to 10000.

PCK frames

• PCK frames are bodyfixed frames. The orientation of a PCK frame is always expressed relative to an inertial frame. The relationship between a PCK frame and its associated inertial frame is provided by a PCK kernel. PCK frames have ID codes between 10000 and 100000. There are two types of PCK kernels---binary and text. Binary PCK kernels are loaded (and unloaded) in a fashion analogous to the loading and unloading of SPK files. To load a binary PCK file

furnsh( <file> )

• To unload a binary PCK file

unload( <file> )

• Text based PCK files are loaded via the function furnsh().

furnsh( <file> )

CK Frames

• CK frames are frames that are defined relative to a spacecraft structure. The orientation of the structure is provided through a binary SPICE kernel called a C-kernel. The ID codes for C-kernel frames are negative and usually less than -999. A C-kernel frame may be defined relative to any other kind of frame. (Most existing C-kernels are defined relative to inertial frames.)

• C-kernels are loaded and unloaded using the same loader functions as used to load and unload SPK kernels. To load a C-kernel

furnsh( <file> )

• To unload a C-kernel

unload( <file> )

• The times used to represent C-kernels are spacecraft clock times---not ET. The relationship between ET and spacecraft clock times is stored in a SPICE text kernel called a spacecraft clock kernel---usually abbreviated as SCLK (ess-clock) kernel. To retrieve states relative to a CK frame you need to make the relationship between ET and the spacecraft clock available to your program by loading the appropriate SCLK kernel. SCLK kernels are loaded via the function furnsh().

furnsh( <sclk_file_name> )

TK Frames

• TK frames (short for Text Kernel frames) are frames that are defined via a SPICE text kernel. These frames can be transformed to another reference frame via a constant rotation matrix. Typical examples are topocentric frames and instrument frames. TK frames are loaded via the function furnsh().

furnsh( <TK_frame_file> )

Dynamic Frames

• Dynamic frames, like TK frames, are defined via a SPICE text kernel. A dynamic frame has time-varying rotation relative to its base frame. A dynamic frame can be defined by two time-varying vectors, by a set of precession, nutation, and obliquity models, or by a set of Euler angles. Typical examples are the geocentric solar ecliptic frame or the Earth true equator and true equinox of date frame. Dynamic frames are loaded via the function furnsh().

furnsh( <Dynamic_frame_file> )

In addition to the files mentioned above, it may be necessary to load a frame definition file along with the one of the SPICE kernels listed above. (If the producer of the file has done his or her homework this step should be unnecessary.) The frame definition file is a SPICE text kernel that specifies the type of the frame, the center of the frame, the name of the frame, and its ID code. (See frames for more details concerning frame definitions.) As is evident from the above discussion, the use of non-inertial frames requires more data management on the part of the user of the SPICE system. However, this data management problem is not a new problem. In previous versions of the SPICE system the same kernels would have been required. Moreover, in previous versions of the SPICE system, you would have been required to perform all non-inertial transformations in your own code. With the inclusion of non-inertial frames in the SPK system, we have relieved you of some of the tasks associated with non-inertial frames.

SPK File Structure

An SPK file is made up of one or more data segments and a comment area. These components are described below.

Segments--The Fundamental SPK Building Blocks

An SPK file contains one or more segments. Each segment contains ephemeris data sufficient to compute the geometric state (position and velocity) of one solar system body (the target) with respect to another (the center) at any epoch throughout some finite interval of time.

Either body may be a spacecraft, a planet or planet barycenter, a satellite, a comet, an asteroid, a tracking station, a roving vehicle, or an arbitrary point for which an ephemeris has been calculated. Each body in the solar system is associated with a unique integer code. A list of names and codes for the planets, major satellites, spacecraft, asteroids and comets can be found in the document naif_ids

The states computed from the ephemeris data in a segment must be referenced to a single, recognized reference frame.

The data in each segment are stored as an array of double precision numbers. The summary for the array, called a descriptor, has two double precision components:

1. The initial epoch of the interval for which ephemeris data are contained in the segment, given in ephemeris seconds past Julian year 2000.

2. The final epoch of the interval for which ephemeris data are contained in the segment, given in ephemeris seconds past Julian year 2000.

The descriptor has six integer components:

1. The NAIF integer code for the target.

2. The NAIF integer code for the center.

3. The NAIF integer code for the reference frame.

4. The integer code for the representation (type of ephemeris data).

5. The initial address of the array.

6. The final address of the array.

In addition to a descriptor, each array also has a name. The name of each array may contain up to 40 characters. This space may be used to store a brief description of the segment. For example, the name may contain pedigree information concerning the segment or may contain the name of the object whose position is recorded in the segment.

Segment Order and Priority

Segments within an SPK file need not be ordered according to time; segments covering (that is, providing data for) a later time period may precede segments covering an earlier time period.

However, segment order does imply priority. For a given target body, segment priority increases with distance from the start of the file: segments closer to the end of the file have higher priority than segments for the same target body that occur earlier in the file. When a data request for a specified target body is made, the segment for that target body with highest priority, and whose time interval includes the request time, will be selected to satisfy the request.

SPK producers should note that this priority scheme would cause a higher priority segment for a target body to mask a lower priority segment for the same body over the intersection of the coverage intervals of the two segments, if two such segments were written to an SPK file. In particular, if an SPK file contained two segments for the same target body and time interval, where the segments had different central bodies, the lower priority segment would be invisible to the SPK system.

The Comment Area

Preceding the segments, the Comment Area provides space in the SPK file for storing textual information besides what is written in the array names. Ideally, each SPK file would contain internal documentation that describes the origin, recommended use, and any other pertinent information about the data in that file. For example, the beginning and ending epochs for the file, the names and NAIF integer codes of the bodies included, an accuracy estimate, the date the file was produced, and the names of the source files used in making the SPK file could be included in the Comment Area.

The utility programs COMMNT and SPACIT may be used to examine and manipulate the comments in an SPK file. In addition to these utilities, CSPICE provides a family of functions for handling this Comment Area. The name of each function in this family begins with the letters SPC which stand for SPk and Ck because this feature is common to both types of files. The SPC software provides the ability to add, extract, read, and delete comments and convert commented files from binary format to SPICE transfer format and back to binary again.

The SPC functions and their functions are described in detail in the SPC Required Reading, spc.

SPK Data Types

The fourth integer component of the descriptor---the code for the representation, or data type ---is the key to the SPK format.

For purposes of determining the segment best suited to fulfill a particular request, all segments are treated equally. It is only when the data in a segment are to be evaluated---when a state vector is to be computed---that the type of data used to represent the ephemeris becomes important.

Because this step is isolated within a single low-level reader, spkpvn(), new data types can be added to the SPK format without affecting application programs that use the higher level readers. spkpvn() is designed so that the changes required to implement a new data type are minimal.

There are no real limits on the possible representations that can be used for ephemeris data. Users with access to data suitable for creating an ephemeris may choose to invent their own representations, adapting spkpvn() accordingly. (We recommend that you consult with NAIF prior to implementing a new data type.)

The data types currently supported by CSPICE software are listed under Supported Data Types later in this document.

Primitive States

At the lowest level, it is possible to compute states without combining them at all. Given the handle and descriptor for a particular segment, function spkpvn() returns a state from that segment directly.

ref, state, center = spkpvn( <handle>, <descr>, <et>)  #  Position, velocity, native frame

spkpvn() is the most basic SPK reader. It returns states relative to the frame in which they are stored in the SPK file. It does not rotate or combine them: it returns a state relative to the center whose integer code is stored in the descriptor for the segment. This state is relative to the frame whose integer ID code is also stored in the descriptor of the segment. The user is responsible for using that state correctly. The user is also responsible for using DAF functions to determine the particular file and segment from which each state is to be computed.

Note that to use the state returned by spkpvn() in any frame other than the native frame of the segment, you must convert the state to the frame of interest.

If files have been loaded by previous calls to furnsh(), it is possible to use the same segments that would normally be used by spkezr(), spkez(), spkssb_c, and spkgeo(). Function spksfs() selects, from the database of loaded files, the file handle and segment descriptor for the segment best suited to the request. If two segments from different files are suitable, spksfs() selects the one from the file that was loaded later. If two segments from the same file are suitable, spksfs() selects the one that is stored later in the file. The call

handle, descr, segnam = spksfs( <801>, <et>, idlen) #  Select file and segment

returns the handle, descriptor, and segment name for the latest segment containing data for Triton at the specified epoch. spksfs() maintains a buffer of segment descriptors and segment names, so it doesn't waste time searching the database for bodies it already knows about.

Examples of Using SPK Readers

Example 1: Computing Latitude and Longitude

The next several sections present sample programs to show how the SPK readers can be used to compute state vectors, and how those vectors can be used to compute derived quantities.

All functions used in the examples are from CSPICE. The convention of expanding SPK function names will be dropped for these examples.

The first example program computes the planetocentric latitude and longitude of the sub-observer point on a target body for any combination of observer, target, and epoch. (Note that planetocentric coordinates differ from planetographic and cartographic coordinates in that they are always right-handed, regardless of the rotation of the body. Also note that for this example we define the sub-observer point to be the point on the surface of the target that lies on the ray from the center of the target to the observer. )

#!/usr/bin/env python3
"""
PROGRAM LATLON
"""

# Standard includes
import spiceypy as spice
import numpy as np

def main():
    # Load constants into the kernel pool. Two files are
    # needed. The first ("leapseconds.ker") contains the dates
    # of leap seconds and values for constants needed to
    # compute the difference between UTC and ET at any
    # epoch. The second ("pck.ker") contains IAU values
    # needed to compute transformations from inertial
    # (J2000) coordinates to body-fixed (pole and prime
    # meridian) coordinates for the major bodies of the
    # solar system.
    spice.furnsh("leapseconds.ker")
    spice.furnsh("pck.ker")

    # Several ephemeris files are used. Most contain data for
    # a single planetary system ("jupiter.bsp", "saturn.bsp",
    # and so on). Some contain data for spacecraft ("vgr1.bsp",
    # "mgn.bsp").
    spice.furnsh("mercury.bsp")
    spice.furnsh("venus.bsp")
    spice.furnsh("earth.bsp")
    spice.furnsh("mars.bsp")
    spice.furnsh("jupiter.bsp")
    spice.furnsh("saturn.bsp")
    spice.furnsh("uranus.bsp")
    spice.furnsh("neptune.bsp")
    spice.furnsh("pluto.bsp")
    spice.furnsh("vgr1.bsp")
    spice.furnsh("vgr2.bsp")
    spice.furnsh("mgn.bsp")
    spice.furnsh("gll.bsp")

    # Inputs are entered interactively. The user enters three
    # items: the name for the observer, the name
    # for the target, and the UTC epoch at which the
    # sub-observer point is to be computed (a free-format string).
    #
    # The epoch must be converted to ephemeris time (ET).
    while True:
        obs = input("Observer? ")
        targ = input("Target? ")
        time_str = input("Epoch? ")

        et = spice.str2et(time_str)
        frame = "IAU_" + targ

        # Compute the true state (corrected for light-time)
        # of the target as seen from the observer at the
        # specified epoch in the target fixed reference frame.
        state, lt = spice.spkezr(targ, et, frame, "LT", obs)

        # We need the vector FROM the target TO the observer
        # to compute latitude and longitude. So negate it.
        state = -np.array(state)

        # Convert from rectangular coordinates to latitude and
        # longitude, then from radians to degrees for output.
        # Note: reclat expects a 3-element vector (position only).
        radius, lon, lat = spice.reclat(state)
        lat_deg = np.degrees(lat)
        lon_deg = np.degrees(lon)

        print("\n"
              "Sub-observer latitude (deg): {}\n"
              "             longitude     : {}\n"
              "\n"
              "Range to target (km)       : {}\n"
              "Light-time (sec)           : {}\n".format(lat_deg, lon_deg, radius, lt))

        # Get the next set of inputs.

if __name__ == "__main__":
    main()

Example 2: Occultation or Transit

The second example determines epochs if one target body (spacecraft, planet, or satellite) is occulted by or in transit across another target body as seen from an observer at a user specified epoch. It is similar in both form and generality to the first example.

#!/usr/bin/env python3
"""
PROGRAM OCCTRN
"""

# Standard includes
import spiceypy as spice
import numpy as np
import math

# CSPICE prototypes and definitions.

def main():
    """
    Constants
    """
    NTARG   = 2

    # Load constants into the kernel pool. Two files are
    # needed. The first ("leapseconds.ker") contains the dates
    # of leap seconds and values for constants needed to
    # compute the difference between UTC and ET at any
    # epoch. The second ("radii.tpc") contains values
    # for the tri-axial ellipsoids used to model the major
    # major bodies of the solar system.
    spice.furnsh("leapseconds.ker")
    spice.furnsh("radii.tpc")

    # Several ephemeris files are needed. Most contain data for
    # a single planetary system ("jupiter.ker", "saturn.ker",
    # and so on). Some contain data for spacecraft ("vgr1.ker",
    # "mgn.ker").
    spice.furnsh("mercury.bsp")
    spice.furnsh("venus.bsp")
    spice.furnsh("earth.bsp")
    spice.furnsh("mars.bsp")
    spice.furnsh("jupiter.bsp")
    spice.furnsh("saturn.bsp")
    spice.furnsh("uranus.bsp")
    spice.furnsh("neptune.bsp")
    spice.furnsh("pluto.bsp")
    spice.furnsh("vgr1.bsp")
    spice.furnsh("vgr2.bsp")
    spice.furnsh("mgn.bsp")
    spice.furnsh("gll.bsp")

    # Inputs are entered interactively. The user enters four
    # items: the code for the observer (an integer), the codes
    # for two target bodies (integers), and the epoch at which
    # check for occultation or transit is to be computed
    # (a free-format string).
    #
    # The epoch must be converted to ephemeris time (ET).
    while True:
        obs = input("Observer? ")
        targ = [None] * NTARG
        targ[0] = input("Target 1? ")
        targ[1] = input("Target 2? ")
        time_str = input("Epoch?    ")

        et = spice.str2et(time_str)

        # Get the ID codes associated with the targets
        # In spiceypy, bodn2c returns the NAIF ID code for a given name.
        try:
            t_ids = [spice.bodn2c(targ[0]), spice.bodn2c(targ[1])]
        except Exception as e:
            print("Error converting body name to ID:", e)
            continue

        d = [0.0, 0.0]
        r = [0.0, 0.0]
        s = [None, None]  # state vectors

        # Get the apparent states of the target objects as seen from
        # the observer. Also get the apparent radius of each object
        # from the kernel pool. (Use zero radius for any spacecraft;
        # use average radius for anything else.)
        #
        # targ[i]          is the ID code of the i'th target, i = 0, 1.
        # state[i][0..5]   is the apparent state of the i'th target.
        #
        # d[i]         is the apparent distance to the i'th target.
        #
        # r[i]         is the apparent radius of the i'th target.
        #
        # Function vnorm_c returns the Euclidean norm (magnitude) of
        # a three-vector.
        #
        # Function sumad_c returns the sum of the elements in a
        # double precision array.
        for i in range(NTARG):
            state, lt = spice.spkezr(targ[i], et, "J2000", "LT+S", obs)
            s[i] = state
            d[i] = np.linalg.norm(state[:3])
            if t_ids[i] < 0:
                r[i] = 0.0
            else:
                radii, dim = spice.bodvcd(t_ids[i], "RADII", 3)
                avg = np.sum(radii) / 3.0
                # Use arcsin to compute the apparent radius
                ratio = avg / d[i]
                # Clamp ratio to valid domain for arcsin
                ratio = max(min(ratio, 1.0), -1.0)
                r[i] = np.arcsin(ratio)

        # Determine the separation between the two bodies. If the
        # separation between the centers is greater than the sum of
        # the apparent radii, then the target bodies are clear of
        # each other.
        #
        # Function vsep_c returns the angle of separation between
        # two three-vectors.
        sep = spice.vsep(s[0][:3], s[1][:3]) - (r[0] + r[1])

        if sep > 0.0:
            print("\nClear.\n")
        else:
            #
            # Otherwise, the smaller body is either occulted or
            # in transit.  We compare ranges to decide which.
            #
            # Let index j indicate the target of smaller radius; let k
            # indicate the larger target.
            j = 0 if r[0] < r[1] else 1
            k = 1 - j

            if d[j] < d[k]:
                print(f"\n{targ[j]} is in transit across {targ[k]}\n")
            else:
                print(f"\n{targ[j]} is occulted by {targ[k]}\n")

        # Get the next set of inputs.

if __name__ == "__main__":
    main()

Additional, working examples of using the principal SPK functions may be found in the Cookbook programs distributed with the SPICE Toolkit.

Supported Data Types

The following representations, or data types, are currently supported by the SPK functions in CSPICE.

1. Modified Difference Arrays.

• Created by the JPL Orbit Determination Program (ODP), these are used primarily for spacecraft ephemerides.

1. Chebyshev polynomials (position only).

• These are sets of coefficients for the x, y, and z components of the body position. The velocity of the body is obtained by differentiation. This data type is normally used for planet barycenters, and for satellites whose orbits are integrated.

1. Chebyshev polynomials (position and velocity).

• These are sets of coefficients for the x, y, and z components of the body position, and for the corresponding components of the velocity. This data type is normally used for satellites whose orbits are computed directly from theories.

1. Reserved for future use (TRW elements for TDRS and Space Telescope).

2. Discrete states (two body propagation).

• This data type contains discrete state vectors. A state is obtained for a specified epoch by propagating the state vectors to that epoch according to the laws of two body motion and then taking a weighted average of the resulting states. Normally, this data type is used for comets and asteroids, whose ephemerides are integrated from an initial state or set of osculating elements.

1. Reserved for future use (Analytic Model for Phobos and Deimos).

2. Reserved for future use (Precessing Classical Elements---used by STScI).

3. Equally spaced discrete states (Lagrange interpolation)

• This data type contains discrete state vectors whose time tags are separated by a constant step size. A state is obtained for a specified epoch by finding a set of states centered at that epoch and using Lagrange interpolation on each component of the states.

1. Unequally spaced discrete states (Lagrange interpolation)

• This data type contains discrete state vectors whose time tags may be unequally spaced. A state is obtained for a specified epoch by finding a set of states centered at that epoch and using Lagrange interpolation on each component of the states.

1. Space Command Two-line Elements

• This data type contains Space Command two-line element representations for objects in Earth orbit (formally called NORAD two-line elements).

1. Reserved for future use.

2. Hermite Interpolation Uniform Spacing.

3. Hermite Interpolation Non-uniform Spacing.

4. Chebyshev polynomials non-uniform spacing (position and velocity).

• This data type contains Chebyshev polynomial coefficients for the position and velocity of an object. Unlike SPK Types 2 and 3, the time intervals to which polynomial coefficient sets apply do not have uniform duration.

1. Precessing conic propagation.

• This data type allows for first order precession of the line of apsides and regression of the line of nodes due to the effects of the J2 coefficient in the harmonic expansion of the gravitational potential of an oblate spheroid.

1. Reserved for future use (Elements for European Space Agency's ISO spacecraft).

2. Equinoctial Elements

• This data type represents the motion of an object about another using equinoctial elements. It provides for precession of the line of apsides and regression of the line of nodes. Unlike Type 15, the mean motion, regression of the nodes and precession of the line of apsides are not derived from the gravitational properties of the central body, but are empirical values.

1. ESOC/DDID Hermite/Lagrange Interpolation

• This data type has been provided to support accurate duplication within the SPK system of spacecraft ephemerides used by the European Space Agency (ESA) on the Mars Express, Rosetta, SMART-1, and Venus express missions.

1. ESOC/DDID Piecewise Interpolation

• SPK type 19 is an enhanced version of SPK type 18. Type 19 enables creation of SPK files representing the same ephemerides that can be represented using type 18, but containing far fewer segments. Data from multiple type 18 segments can be stored in a single type 19 segment.

1. Chebyshev (velocity only)

• SPK data type 20 contains Chebyshev polynomial coefficients for the velocity of a body, relative to its center of motion, as a function of time. The position of the body is obtained by integrating the velocity using a specified integration constant. This data type is provided to accurately represent EPM ephemerides developed by the Institute of Applied Astronomy (IAA), Russian Academy of Sciences (RAS).

1. Extended Modified Difference Arrays

• SPK data type 21 contains extended Modified Difference Arrays (MDA), also called difference lines. These data structures use the same mathematical trajectory representation as SPK data type 1, but type 21 allows use of larger, higher-degree MDAs.

Because SPK files are Double Precision Array Files (DAFs), each segment is stored as an array. Each array corresponding to a particular data type has a particular internal structure. These structures (for the non-reserved types) are described below.

Type 1: Modified Difference Arrays

The first SPK data type contains Modified Difference Arrays (MDA), sometimes called difference lines. This data type is normally used for spacecraft whose ephemerides are produced by JPL's principal trajectory integrator---DPTRAJ. Difference lines are extracted from the spacecraft trajectory file (P-files and PV-files) created by DPTRAJ.

Each segment containing Modified Difference Arrays contains an arbitrary number of logical records. Each record contains difference line coefficients valid up to some final epoch, along with the state at that epoch. The contents of the records themselves are described in [163]. The function spke01_ contains the algorithm used to construct a state from a particular record and epoch.

The records within a segment are ordered by increasing final epoch. The final epochs associated with the records must be distinct.

A segment of this type is structured as follows:

+-----------------------------------------+
| Record 1 (difference line coefficients) |
+-----------------------------------------+
| Record 2 (difference line coefficients) |
+-----------------------------------------+
  .
  .
  .
+-----------------------------------------+
| Record N (difference line coefficients) |
+-----------------------------------------+
| Epoch 1                      |
+------------------------------+
| Epoch 2                      |
+------------------------------+
  .
  .
  .
+------------------------------+
| Epoch N                      |
+------------------------------+
| Epoch 100                    |   (First directory epoch)
+------------------------------+
| Epoch 200                    |   (Second directory epoch)
+------------------------------+
  .
  .
  .
+------------------------------+
| Epoch (N/100)*100            |   (Final directory epoch)
+------------------------------+
| N                            |
+------------------------------+

The number of records in a segment, N, can be arbitrarily large. Records 1 through N contain the difference line coefficients and other constants needed to compute state data. Each one of these records contains 71 double precision numbers.

The list of final epochs for the records is stored immediately after the last record.

Following the list of epochs is a second list, the directory, containing every 100th epoch from the previous list. If there are N epochs, there will be N/100 directory epochs. If there are fewer than 100 epochs, then the segment will not contain any directory epochs. Directory epochs are used to speed up access to desired records.

The final element in the segment is the number of records contained in the segment, N.

The index of the record corresponding to a particular epoch is the index of the first epoch not less than the target epoch.

Type 2: Chebyshev (position only)

The second SPK data type contains Chebyshev polynomial coefficients for the position of the body as a function of time. Normally, this data type is used for planet barycenters, and for satellites whose ephemerides are integrated. (The velocity of the body is obtained by differentiating the position.)

Each segment contains an arbitrary number of logical records. Each record contains a set of Chebyshev coefficients valid throughout an interval of fixed length. The function spke02_ contains the algorithm used to construct a state from a particular record and epoch.

The records within a segment are ordered by increasing initial epoch. All records contain the same number of coefficients. A segment of this type is structured as follows:

+---------------+
| Record 1      |
+---------------+
| Record 2      |
+---------------+
  .
  .
  .
+---------------+
| Record N      |
+---------------+
| INIT          |
+---------------+
| INTLEN        |
+---------------+
| RSIZE         |
+---------------+
| N             |
+---------------+

A four-number directory at the end of the segment contains the information needed to determine the location of the record corresponding to a particular epoch.

1. INIT is the initial epoch of the first record, given in ephemeris seconds past J2000.

2. INTLEN is the length of the interval covered by each record, in seconds.

3. RSIZE is the total size of (number of array elements in) each record.

4. N is the number of records contained in the segment.

Each record is structured as follows:

+------------------+
| MID              |
+------------------+
| RADIUS           |
+------------------+
| X  coefficients  |
+------------------+
| Y  coefficients  |
+------------------+
| Z  coefficients  |
+------------------+

The first two elements in the record, MID and RADIUS, are the midpoint and radius of the time interval covered by coefficients in the record. These are used as parameters to perform transformations between the domain of the record (from MID - RADIUS to MID + RADIUS) and the domain of Chebyshev polynomials (from -1 to 1 ). The same number of coefficients is always used for each component, and all records are the same size (RSIZE), so the degree of each polynomial is

( RSIZE - 2 ) / 3 - 1

To facilitate the creation of Type 2 segments, a segment writing function called spkw02() has been provided. This function takes as input arguments the handle of an SPK file that is open for writing, the information needed to construct the segment descriptor, and the data to be stored in the segment. The header of the function provides a complete description of the input arguments and an example of its usage.

Type 3: Chebyshev (position and velocity)

The third SPK data type contains Chebyshev polynomial coefficients for the position and velocity of the body as a function of time. Normally, this data type is used for satellites for which the ephemerides are computed from analytical theories.

The structure of the segment is nearly identical to that of the SPK data Type 2 (Chebyshev polynomials for position only). The only difference is that each logical record contains six sets of coefficients instead of three. The function spke03_ contains the algorithm used to construct a state from a particular record and epoch.

Each record is structured as follows:

+------------------+
| MID              |
+------------------+
| RADIUS           |
+------------------+
| X  coefficients  |
+------------------+
| Y  coefficients  |
+------------------+
| Z  coefficients  |
+------------------+
| X' coefficients  |
+------------------+
| Y' coefficients  |
+------------------+
| Z' coefficients  |
+------------------+

The same number of coefficients is always used for each component, and all records are the same size (RSIZE), so the degree of each polynomial is

( RSIZE - 2 ) / 6 - 1

To facilitate the creation of Type 3 segments, a segment writing function called spkw03() has been provided. This function takes as input arguments the handle of an SPK file that is open for writing, the information needed to construct the segment descriptor, and the data to be stored in the segment. The header of the function provides a complete description of the input arguments and an example of its usage.

Type 5: Discrete states (two body propagation)

The fifth standard SPK data type contains discrete state vectors. A state is obtained from a Type 5 segment for any epoch that is within the bounds of that segment by propagating the discrete states to the specified epoch according to the laws of two body motion. Normally, this data type is used for comets and asteroids, whose ephemerides are integrated from an initial state or set of osculating elements.

Each segment contains of a number of logical records. Each record consists of an epoch (ephemeris seconds past J2000) and the geometric state of the body at that epoch (x, y, z, dx/dt, dy/dt, dz/dt, in kilometers and kilometers per second). Records are ordered with respect to increasing time.

The records that correspond to an epoch for which a state is desired are the ones whose associated epochs bracket that epoch. The state in each record is used as the initial state in a two-body propagation; a weighted average of the propagated states gives the position of the body at the specified epoch. The velocity is given by the derivative of the position. Thus the position and velocity at the specified epoch are given by:

 P  = W(t) * P1(t) + (1-W(t)) * P2(t)


V  = W(t) * V1(t) + (1-W(t)) * V2(t) + W'(t) * ( P1(t) - P2(t) )

where P1, V1, P2, and V2 are the position and velocity components of the propagated states and W is the weighting function. The weighting function used is:

W(t) = 0.5 + 0.5 * cos [ PI * ( t - t1 ) / ( t2 - t1 ) ]

where t1 and t2 are the epochs that bracket the specified epoch t. Physically, the epochs and states are stored separately, so that the epochs can be searched as an ordered array. Thus, the initial part of each segment looks like this:

+--------------------+
| State 1            |
+--------------------+
         .
         .
         .
+--------------------+
| State N            |
+--------------------+
| Epoch 1            |
+--------------------+
         .
         .
         .
+--------------------+
| Epoch N            |
+--------------------+

The number of records in a segment can be arbitrarily large. In order to avoid the file reads required to search through a large array of epochs, each segment contains a simple directory immediately after the final epoch. This directory contains every 100th epoch in the epoch array. If there are N epochs, there will be N/100 directory epochs. (If there are fewer than 100 epochs, no directory epochs are stored.)

The final items in the segment are GM, the gravitational parameter of the central body (kilometers and seconds), and N, the number of states in the segment. Thus, the complete segment looks like this:

+--------------------+
| State 1            |
+--------------------+
         .
         .
         .
+--------------------+
| Epoch 1            |
+--------------------+
         .
         .
         .
+--------------------+
| Epoch N            |
+--------------------+
| Epoch 100          |           (First directory epoch)
+--------------------+
| Epoch 200          |
+--------------------+
         .
         .
         .
+--------------------+
| Epoch (N/100)*100  |           (Final directory epoch)
+--------------------+
| GM                 |
+--------------------+
| N                  |
+--------------------+

To facilitate the creation of Type 5 segments, a segment writing function called spkw05() has been provided. This function takes as input arguments the handle of an SPK file that is open for writing, the information needed to construct the segment descriptor, and the data to be stored in the segment. The header of the function provides a complete description of the input arguments and an example of its usage.

Type 8: Lagrange Interpolation --- Equal Time Steps

The eighth SPK data type represents a continuous ephemeris using a discrete set of states and a Lagrange interpolation method. The epochs (also called time tags) associated with the states must be evenly spaced: there must be some positive constant STEP such that each time tag differs from its predecessor and successor by STEP seconds. For a request epoch not corresponding to the time tag of some state, the data type defines a state by interpolating each component of a set of states whose epochs are centered near the request epoch. Details of how these states are selected and interpolated are given below.

The SPK system can also represent an ephemeris using unequally spaced discrete states and Lagrange interpolation; SPK Type 9 does this. SPK Type 9 sacrifices some run-time speed and economy of storage in order to achieve greater flexibility.

The states in a Type 8 segment are geometric: they do not take into account aberration corrections. The six components of each state vector represent the position and velocity (x, y, z, dx/dt, dy/dt, dz/dt, in kilometers and kilometers per second) of the body to which the ephemeris applies, relative to the center specified by the segment's descriptor. The epochs corresponding to the states are barycentric dynamical times (TDB), expressed as seconds past J2000.

Each segment also has a polynomial degree associated with it; this is the degree of the interpolating polynomials to be used in evaluating states based on the data in the segment. The identical degree is used for interpolation of each state component.

Type 8 SPK segments have the structure shown below:

                                    +--------+
                                    |  x(1)  |
                                /   +--------+
                               /    |  y(1)  |
                              /     +--------+
                             /      |  z(1)  |
+-----------------------+   /       +--------+
| State 1               |  <        |dx(1)/dt|
+-----------------------+   \       +--------+
| State 2               |    \      |dy(1)/dt|
+-----------------------+     \     +--------+
            .                  \    |dz(1)/dt|
            .                       +--------+
            .
+-----------------------+
| State N               |
+-----------------------+
| Epoch of state 1 (TDB)|
+-----------------------+
| Step size             |
+-----------------------+
| Polynomial degree     |
+-----------------------+
| Number of states      |
+-----------------------+

In the diagram, each box representing a state vector corresponds to six double precision numbers; the other boxes represent individual double precision numbers. Since the epochs of the states are evenly spaced, they are represented by a start epoch and a step size. The number of states must be greater than the interpolating polynomial degree. The Type 8 interpolation method works as follows: given an epoch at which a state is requested and a segment having coverage for that epoch, the Type 8 reader finds a group of states whose epochs are centered about the epoch. The size of the group is one greater than the polynomial degree associated with the segment. If the group size N is even, then the group will consist of N consecutive states such that the request time is between the epochs of the members of the group having indices, relative to the start of the group, of N/2 and (N/2 + 1), inclusive. When N is odd, the group will contain a central state whose epoch is closest to the request time, and will also contain (N-1)/2 neighboring states on either side of the central one. The Type 8 evaluator will then use Lagrange interpolation on each component of the states to produce a state corresponding to the request time. For the jth state component, the interpolation algorithm is mathematically equivalent to finding the unique polynomial of degree N-1 that interpolates the ordered pairs

( epoch(i), state(j,i) ),  i = k ,  k , ... , k
                                1    2         N

and evaluating the polynomial at the requested epoch. Here

k ,  k , ... , k
 1    2         N

are the indices of the states in the interpolation group,

epoch(i)

is the epoch of the ith state and

state(j,i)

is the jth component of the ith state. There is an exception to the state selection algorithm described above: the request time may be too near the first or last state of the segment to be properly bracketed. In this case, the set of states selected for interpolation still has size N, and includes either the first or last state of the segment.

To facilitate the creation of Type 8 segments, a segment writing function called spkw08() has been provided. This function takes as input arguments the handle of an SPK file that is open for writing, the information needed to construct the segment descriptor, and the data to be stored in the segment. The header of the function provides a complete description of the input arguments and an example of its usage.

Type 9: Lagrange Interpolation --- Unequal Time Steps

The ninth SPK data type represents a continuous ephemeris using a discrete set of states and a Lagrange interpolation method. The epochs (also called time tags ) associated with the states need not be evenly spaced. For a request epoch not corresponding to the time tag of some state, the data type defines a state by interpolating each component of a set of states whose epochs are centered near the request epoch. Details of how these states are selected and interpolated are given below.

The states in a Type 9 segment are geometric: they do not take into account aberration corrections. The six components of each state vector represent the position and velocity (x, y, z, dx/dt, dy/dt, dz/dt, in kilometers and kilometers per second) of the body to which the ephemeris applies, relative to the center specified by the segment's descriptor. The epochs corresponding to the states are barycentric dynamical times (TDB), expressed as seconds past J2000.

Each segment also has a polynomial degree associated with it; this is the degree of the interpolating polynomials to be used in evaluating states based on the data in the segment. The identical degree is used for interpolation of each state component.

Type 9 SPK segments have the structure shown below:

                                    +--------+
                                    |  x(1)  |
                                /   +--------+
                               /    |  y(1)  |
                              /     +--------+
                             /      |  z(1)  |
+-----------------------+   /       +--------+
| State 1               |  <        |dx(1)/dt|
+-----------------------+   \       +--------+
| State 2               |    \      |dy(1)/dt|
+-----------------------+     \     +--------+
            .                  \    |dz(1)/dt|
            .                       +--------+
            .
+-----------------------+
| State N               |
+-----------------------+
| Epoch 1               |
+-----------------------+
| Epoch 2               |
+-----------------------+
            .
            .
            .
+-----------------------+
| Epoch N               |
+-----------------------+
| Epoch 100             | (First directory)
+-----------------------+
            .
            .
            .
+-----------------------+
| Epoch ((N-1)/100)*100 | (Last directory)
+-----------------------+
| Polynomial degree     |
+-----------------------+
| Number of states      |
+-----------------------+

In the diagram, each box representing a state vector corresponds to six double precision numbers; the other boxes represent individual double precision numbers. The number of states must be greater than the interpolating polynomial degree. The set of time tags is augmented by a series of directory entries; these entries allow the Type 9 reader to search for states more efficiently. The directory entries contain time tags whose indices are multiples of 100. The set of indices of time tags stored in the directories ranges from 100 to

(  (N-1) / 100  ) * 100

where N is the total number of time tags. Note that if N is

Q * 100

then only

Q - 1

directory entries are stored, and in particular, if there are only 100 states in the segment, there are no directories. The Type 9 interpolation algorithm is virtually identical to the Type 8 algorithm; see the discussion of SPK Type 8 for details. However, the Type 9 algorithm executes more slowly than the Type 8 algorithm, since the Type 9 reader must search through tables of time tags to find appropriates states to interpolate, while the Type 8 reader can locate the correct set of states to interpolate by a direct computation.

To facilitate the creation of Type 9 segments, a segment writing function called spkw09() has been provided. This function takes as input arguments the handle of an SPK file that is open for writing, the information needed to construct the segment descriptor, and the data to be stored in the segment. The header of the function provides a complete description of the input arguments and an example of its usage.

Type 10: Space Command Two-Line Elements

The SPK data Type 10 uses the SPICE concept of a generic segment to store a collection of packets each of which models the trajectory of some Earth satellite using Space Command two-line element sets (TLEs) (formerly the North American Air Defense --- NORAD). TLE propagation occurs using the algorithms as described in the Spacetrak 3 report for SGP4 and SDP4. Note: The Spacetrak 3 implementation of SDP4 contained several programming errors. The errors were corrected for CSPICE implementation.

The SPICE generic segment software handles storage, arrangement, and retrieval of the TLEs. We review only the pertinent points about generic segments here.

A generic SPK segment contains several logical data partitions:

1. A partition for constant values to be associated with each data packet in the segment.

2. A partition for the data packets.

3. A partition for epochs.

4. A partition for a packet directory, if the segment contains variable sized packets.

5. A partition for an epoch directory.

6. A reserved partition that is not currently used. This partition is only for the use of the NAIF group at the Jet Propulsion Laboratory (JPL).

7. A partition for the meta data which describes the locations and sizes of other partitions as well as providing some additional descriptive information about the generic segment.

+============================+
|         Constants          |
+============================+
|          Packet 1          |
|----------------------------|
|          Packet 2          |
|----------------------------|
|              .             |
|              .             |
|              .             |
|----------------------------|
|          Packet N          |
+============================+
|      Reference Epochs      |
+============================+
|      Packet Directory      |
+============================+
|       Epoch Directory      |
+============================+
|       Reserved  Area       |
+============================+
|     Segment Meta Data      |
+----------------------------+

Each packet of a Type 10 segment contains a set of two-line elements, the nutations in longitude and obliquity of the Earth's pole, and the rates of these nutations. Each packet is arranged as shown below. (The notation below is taken from the description that accompanies the code available from Space Command for the evaluation of two-line elements.)

   A single SPK Type 10 segment packet

   +-------------------+
 1 |      NDT20        |
   +-------------------+
 2 |      NDD60        |
   +-------------------+
 3 |      BSTAR        |
   +-------------------+
 4 |      INCL         |
   +-------------------+
 5 |      NODE0        |     Two-line element packet
   +-------------------+
 6 |      ECC          |
   +-------------------+
 7 |      OMEGA        |
   +-------------------+
 8 |      MO           |
   +-------------------+
 9 |      NO           |
   +-------------------+
10 |      EPOCH        |
   +-------------------+
11 |      NU.OBLIQUITY |
   +-------------------+
12 |      NU.LONGITUDE |
   +-------------------+
13 |     dOBLIQUITY/dt |
   +-------------------+
14 |     dLONGITUDE/dt |
   +-------------------+

The constants partition of the Type 10 segment contains the following eight geophysical constants.

   +-------------------------------------------+
1  |  J2 gravitational harmonic for Earth      |
   +-------------------------------------------+
2  |  J3 gravitational harmonic for Earth      |
   +-------------------------------------------+
3  |  J4 gravitational harmonic for Earth      |
   +-------------------------------------------+
   |  Square root of the GM for Earth where GM |
4  |  is expressed in Earth radii cubed per    |
   |  minutes squared                          |
   +-------------------------------------------+
5  |  High altitude bound for atmospheric      |
   |  model in km                              |
   +-------------------------------------------+
6  |  Low altitude bound for atmospheric       |
   |  model in km                              |
   +-------------------------------------------+
7  |  Equatorial radius of the Earth in km     |
   +-------------------------------------------+
8  |  Distance units/Earth radius (normally 1) |
   +-------------------------------------------+

The reference epochs partition contains an ordered collection of epochs. The i'th reference epoch is equal to the epoch in the i'th packet. The epoch directory contains every 100th reference epoch. The epoch directory is used to efficiently locate an the reference epoch that should be associated with a two line element packet.

The packet directory is empty.

Access to the data should be made via the SPK Type 10 reader---spkr10_ or via the SPICELIB generic segment functions. Use the function spkw10() to write a Type 10 generic segment.

Type 12: Hermite Interpolation --- Equal Time Steps

The twelfth SPK data type represents a continuous ephemeris using a discrete set of states and a sliding window Hermite interpolation method. The epochs, also called "time tags," associated with the states must be evenly spaced: there must be some positive constant STEP such that each time tag differs from its predecessor by STEP seconds. For any request epoch, the data type defines a state by interpolating a set of consecutive states, or "window," centered as closely as possible to the request epoch. Interpolated position values are obtained for each coordinate by fitting a Hermite polynomial to the window's set of position and velocity values for that coordinate; interpolated velocity is obtained by differentiating the interpolating polynomials. Details of the interpolation method are given below.

The SPK system can also represent an ephemeris using unequally spaced discrete states and Hermite interpolation; SPK type 13 does this. SPK type 13 sacrifices some run-time speed and economy of storage in order to achieve greater flexibility.

The states in a type 12 segment are geometric: they do not take into account aberration corrections. The six components of each state vector represent the position and velocity (x, y, z, dx/dt, dy/dt, dz/dt, in kilometers and kilometers per second) of the body to which the ephemeris applies, relative to the center specified by the segment's descriptor. The epochs corresponding to the states are barycentric dynamical times (TDB), expressed as seconds past J2000.

Each segment also has a polynomial degree associated with it; this is the degree of the interpolating polynomials to be used in evaluating states based on the data in the segment. The identical degree is used for interpolation of each state component.

Type 12 SPK segments have the structure shown below:

                                    +--------+
                                    |  x(1)  |
                                /   +--------+
                               /    |  y(1)  |
                              /     +--------+
                             /      |  z(1)  |
+-----------------------+   /       +--------+
| State 1               |  <        |dx(1)/dt|
+-----------------------+   \       +--------+
| State 2               |    \      |dy(1)/dt|
+-----------------------+     \     +--------+
            .                  \    |dz(1)/dt|
            .                       +--------+
            .
+-----------------------+
| State N               |
+-----------------------+
| Epoch of state 1 (TDB)|
+-----------------------+
| Step size             |
+-----------------------+
| Window size - 1       |
+-----------------------+
| Number of states      |
+-----------------------+

In the diagram, each box representing a state vector corresponds to six double precision numbers; the other boxes represent individual double precision numbers. Since the epochs of the states are evenly spaced, they are represented by a start epoch and a step size. The number of states must be greater than or equal to the window size, which is related to the polynomial degree as shown:

DEGREE  =  2 * WINDOW_SIZE  -  1

The type 12 interpolation method works as follows: given an epoch at which a state is requested and a segment having coverage for that epoch, the type 12 reader finds a window of states whose epochs are "centered" about the epoch. If the window size S is even, then the window will consist of S consecutive states such that the request time is between the epochs of the members of the group having indices, relative to the start of the group, of S/2 and (S/2 + 1), inclusive. When S is odd, the group will contain a central state whose epoch is closest to the request time, and will also contain (S-1)/2 neighboring states on either side of the central one. For each of the x-, y-, and z-coordinates, the type 12 evaluator will fit an Hermite polynomial to the corresponding position and velocity values of the states in the selected window. Each polynomial is evaluated at the request time to yield the interpolated position components. The derivatives of these polynomials are evaluated at the request time to yield the interpolated velocity components. For the jth coordinate, the interpolation algorithm is mathematically equivalent to finding the unique polynomial of degree 2*S-1 that interpolates the ordered pairs

( epoch(i), position(j,i) ),  i = k ,  k , ... , k
                                   1    2         S

and whose derivative interpolates the ordered pairs

( epoch(i), velocity(j,i) ),  i = k ,  k , ... , k
                                   1    2         S

and evaluating the polynomial and its derivative at the requested epoch. Here

k ,  k , ... , k
 1    2         S

are the indices of the states in the interpolation window,

epoch(i)

is the epoch of the ith state and

position(j,i)
velocity(j,i)

are, respectively, the jth components of the position and velocity comprising the ith state. There is an exception to the state selection algorithm described above: the request time may be too near the first or last state of the segment to be properly bracketed. In this case, the set of states selected for interpolation still has size S, and includes either the first or last state of the segment.

To facilitate the creation of type 12 segments, a segment writing routine called spkw12() has been provided. This routine takes as input arguments the handle of an SPK file that is open for writing, the information needed to construct the segment descriptor, and the data to be stored in the segment. The header of the subroutine provides a complete description of the input arguments and an example of its usage.

Type 13: Hermite Interpolation --- Unequal Time Steps

The thirteenth SPK data type represents a continuous ephemeris using a discrete set of states and a sliding window Hermite interpolation method. The epochs, also called "time tags," associated with the states need not be evenly spaced. For any request epoch, the data type defines a state by interpolating a set of consecutive states, or "window," centered as closely as possible to the request epoch. Interpolated position values are obtained for each coordinate by fitting a Hermite polynomial to the window's set of position and velocity values for that coordinate; interpolated velocity is obtained by differentiating the interpolating polynomials. Details of the interpolation method are given below.

The states in a type 13 segment are geometric: they do not take into account aberration corrections. The six components of each state vector represent the position and velocity (x, y, z, dx/dt, dy/dt, dz/dt, in kilometers and kilometers per second) of the body to which the ephemeris applies, relative to the center specified by the segment's descriptor. The epochs corresponding to the states are barycentric dynamical times (TDB), expressed as seconds past J2000.

Each segment also has a polynomial degree associated with it; this is the degree of the interpolating polynomials to be used in evaluating states based on the data in the segment. The identical degree is used for interpolation of each state component.

Type 13 SPK segments have the structure shown below:

                                    +--------+
                                    |  x(1)  |
                                /   +--------+
                               /    |  y(1)  |
                              /     +--------+
                             /      |  z(1)  |
+-----------------------+   /       +--------+
| State 1               |  <        |dx(1)/dt|
+-----------------------+   \       +--------+
| State 2               |    \      |dy(1)/dt|
+-----------------------+     \     +--------+
            .                  \    |dz(1)/dt|
            .                       +--------+
            .
+-----------------------+
| State N               |
+-----------------------+
| Epoch 1               |
+-----------------------+
| Epoch 2               |
+-----------------------+
            .
            .
            .
+-----------------------+
| Epoch N               |
+-----------------------+
| Epoch 100             | (First directory)
+-----------------------+
            .
            .
            .
+-----------------------+
| Epoch ((N-1)/100)*100 | (Last directory)
+-----------------------+
| Window size - 1       |
+-----------------------+
| Number of states      |
+-----------------------+

In the diagram, each box representing a state vector corresponds to six double precision numbers; the other boxes represent individual double precision numbers. The number of states must be greater than or equal to the window size, which is related to the polynomial degree as shown:

DEGREE  =  2 * WINDOW_SIZE  -  1

The set of time tags is augmented by a series of directory entries; these entries allow the type 13 reader to search for states more efficiently. The directory entries contain time tags whose indices are multiples of 100. The set of indices of time tags stored in the directories ranges from 100 to

(  (N-1) / 100  ) * 100

where N is the total number of time tags. Note that if N is

Q * 100

then only

Q - 1

directory entries are stored, and in particular, if there are only 100 states in the segment, there are no directories. The type 13 interpolation algorithm is virtually identical to the type 12 algorithm; see the discussion of SPK type 12 for details. However, the type 13 algorithm executes more slowly than the type 12 algorithm, since the type 13 reader must search through tables of time tags to find appropriates states to interpolate, while the type 12 reader can locate the correct set of states to interpolate by a direct computation.

To facilitate the creation of type 13 segments, a segment writing routine called spkw13() has been provided. This routine takes as input arguments the handle of an SPK file that is open for writing, the information needed to construct the segment descriptor, and the data to be stored in the segment. The header of the subroutine provides a complete description of the input arguments and an example of its usage.

Type 14: Chebyshev Polynomials --- Unequal Time Steps

The SPK data Type 14 uses the SPICE concept of a generic segment to store a collection of packets each of which models the trajectory of some object with respect to another over some interval of time. Each packet contains a set of coefficients for Chebyshev polynomials that approximate the position and velocity of some object. The time intervals corresponding to each packet are non-overlapping. Moreover their union covers the interval of time spanned by the start and end times of the Type 14 segment. Unlike Types 2 and 3 the time spacing between sets of coefficients for a Type 14 segment may be non-uniform.

The storage, arrangement and retrieval of packets is handled by the SPICE generic segment software. We only review the pertinent points about generic segments here.

A generic SPK segment contains several logical data partitions:

1. A partition for constant values to be associated with each data packet in the segment.

2. A partition for the data packets.

3. A partition for epochs.

4. A partition for a packet directory, if the segment contains variable sized packets.

5. A partition for an epoch directory.

6. A reserved partition that is not currently used. This partition is only for the use of the NAIF group at the Jet Propulsion Laboratory (JPL).

7. A partition for the meta data which describes the locations and sizes of other partitions as well as providing some additional descriptive information about the generic segment.

+============================+
|         Constants          |
+============================+
|          Packet 1          |
|----------------------------|
|          Packet 2          |
|----------------------------|
|              .             |
|              .             |
|              .             |
|----------------------------|
|          Packet N          |
+============================+
|      Reference Epochs      |
+============================+
|      Packet Directory      |
+============================+
|       Epoch Directory      |
+============================+
|       Reserved  Area       |
+============================+
|     Segment Meta Data      |
+----------------------------+

Only the placement of the meta data at the end of a generic segment is required. The other data partitions may occur in any order in the generic segment because the meta data will contain pointers to their appropriate locations within the generic segment. In the case of Type 14 SPK segments each packet contains an epoch, EPOCH, an allowed time offset, OFFSET, from the epoch, and 6 sets of Chebyshev polynomial coefficients which are used to evaluate the x,y,z, dx/dt, dy/dt, and dz/dt components of the state for epochs within OFFSET seconds of the EPOCH. Each packet is organized with the following structure:

------------------------------------------------
|  The midpoint of the approximation interval  |
------------------------------------------------
|  The radius of the approximation interval    |
------------------------------------------------
|  CHBDEG+1 coefficients for the X coordinate  |
------------------------------------------------
|  CHBDEG+1 coefficients for the Y coordinate  |
------------------------------------------------
|  CHBDEG+1 coefficients for the Z coordinate  |
------------------------------------------------
|  CHBDEG+1 coefficients for the X velocity    |
------------------------------------------------
|  CHBDEG+1 coefficients for the Y velocity    |
------------------------------------------------
|  CHBDEG+1 coefficients for the Z velocity    |
------------------------------------------------

The maximum degree Chebyshev representation that can currently be accommodated is 18. Packets are stored in increasing order of the midpoint of the approximation interval. The constants partition contains a single value, the degree of the Chebyshev representation.

The reference epochs partition contains an ordered collection of epochs. The i'th reference epoch corresponds to the beginning of the interval for which the i'th packet can be used to determine the state of the object modeled by this segment.

The epoch directory contains every 100th reference epoch. The epoch directory is used to efficiently locate an the reference epoch that should be associated with an epoch for which a state has been requested.

The packet directory is empty.

As noted above the exact location of the various partitions must be obtained from the Meta data contained at the end of the segment.

Access to the data should be made via the CSPICE generic segment functions.

Type 14 segments should be created using the functions spk14b(), spk14a(), and spk14e(). The usage of these functions is discussed in spk14b().

Type 15: Precessing Conic Propagation

The SPK data Type 15 represents a continuous ephemeris using a compact analytic model. The object is modeled as orbiting a central body under the influence of a central mass plus first order secular effects of the J2 term in harmonic expansion of the central body gravitational potential.

Type 15 SPK segments have the structure shown below:

+--------------------------------+
| Epoch of Periapsis             |
+--------------------------------+
| Trajectory pole_x              |
+--------------------------------+
| Trajectory pole_y              |
+--------------------------------+
| Trajectory pole_z              |
+--------------------------------+
| Periapsis Unit Vector_x        |
+--------------------------------+
| Periapsis Unit Vector_y        |
+--------------------------------+
| Periapsis Unit Vector_z        |
+--------------------------------+
| Semi-Latus Rectum              |
+--------------------------------+
| Eccentricity                   |
+--------------------------------+
| J2 Processing Flag             |
+--------------------------------+
| Central Body Pole_x            |
+--------------------------------+
| Central Body Pole_y            |
+--------------------------------+
| Central Body Pole_z            |
+--------------------------------+
| Central Body GM                |
+--------------------------------+
| Central Body J2                |
+--------------------------------+
| Central Body Equatorial Radius |
+--------------------------------+

It is important to note that the epoch must be that of periapsis passage. Precession of the line of apsides and regression of the line of nodes is computed relative to this epoch. The effects of the J2 term are not applied if the eccentricity is greater than or equal to 1.

To facilitate the creation of Type 15 segments, a segment writing function called spkw15() has been provided. This function takes as input arguments the handle of an SPK file that is open for writing, the information needed to construct the segment descriptor, and the data to be stored in the segment. The header of the function provides a complete description of the input arguments and an example of its usage.

Type 17: Equinoctial Elements

The SPK data Type 17 represents a continuous ephemeris using a compact analytic model. The object is following an elliptic orbit with precessing line of nodes and argument of periapse relative to the equatorial frame of some central body. The orbit is modeled via equinoctial elements.

Type 17 SPK segments have the structure shown below:

    +----------------------------------+
 1  | Epoch of Periapsis               |
    +----------------------------------+
 2  | Semi-Major Axis                  |
    +----------------------------------+
 3  | H term of equinoctial elements   |
    +----------------------------------+
 4  | K term of equinoctial elements   |
    +----------------------------------+
 5  | Mean longitude at epoch          |
    +----------------------------------+
 6  | P term of equinoctial elements   |
    +----------------------------------+
 7  | Q term of equinoctial elements   |
    +----------------------------------+
 8  | rate of longitude of periapse    |
    +----------------------------------+
 9  | mean longitude rate              |
    +----------------------------------+
10  | longitude of ascending node rate |
    +----------------------------------+
11  | equatorial pole right ascension  |
    +----------------------------------+
12  | equatorial pole declination      |
    +----------------------------------+

To facilitate the creation of Type 17 segments, a segment writing function called spkw17() has been provided. This function takes as input arguments the handle of an SPK file that is open for writing, the information needed to construct the segment descriptor, and the data to be stored in the segment. The header of the function provides a complete description of the input arguments and an example of its usage.

Type 18: ESOC/DDID Hermite/Lagrange Interpolation

SPK type 18 has been provided to support accurate duplication within the SPK system of spacecraft ephemerides used by the European Space Agency (ESA) on the Mars Express, Rosetta, SMART-1 and Venus Express missions. However, the algorithms used by SPK type 18 are very general; type 18's applicability is by no means limited to these missions.

Because of the possibility of evolution of the mathematical representations of ephemerides used by ESA, SPK type 18 is designed to accommodate multiple representations, thereby avoiding a proliferation of SPK data types. SPK type 18 refers to each supported mathematical representation of ephemeris data as a subtype.

Currently SPK type 18 supports two subtypes:

1. Subtype 0

• Separate sliding-window Hermite interpolation of position and velocity. The ephemeris is represented by a series of 12-element packets and associated time tags. The time tags may be unequally spaced. Each packet contains three Cartesian position components, three velocity components meant to be used for Hermite interpolation of the position, three velocity components (not necessarily equal to the previous three), and three acceleration components meant to be used with the second set of velocity components for Hermite interpolation of the velocity. The position and velocity resulting from this interpolation method are in principle independent. The same interpolation degree is used for each position and velocity component.

1. Subtype 1

• Separate sliding-window Lagrange interpolation of position and velocity. The ephemeris is represented by a series of 6-element packets and associated time tags. The time tags may be unequally spaced. Each packet contains three Cartesian position components and three velocity components. The position components and velocity components are interpolated separately. The position and velocity resulting from this interpolation method are in principle independent. The same interpolation degree is used for each position and velocity component.

The sliding-window interpolation technique used by this data type works as follows: for any request epoch, the data type defines a component of position or velocity by interpolating a set of values of that component defined on a set of consecutive time tags---a "window"---centered as closely as possible to the request epoch. The nominal window size is dictated by the degree and type (Hermite vs. Lagrange) of the interpolating polynomials. Normally the window of time tags has even size, and the window is selected so that the request time is located between the two central time tags in the window. When the request time is near a segment boundary, the window is truncated if necessary on the side closest to the boundary. If a segment contains too few packets to form a window of nominal size, as many packets as are needed and available are used to construct the window. In this case the window size may be odd. In any case the window never includes more than WNDSIZ/2 time tags on either side of the request time, where WNDSIZ is the nominal window size.

The states in a type 18 segment are geometric: they do not take into account aberration corrections. The position and velocity components of each packet represent the position (x, y, z, in kilometers and kilometers per second) of the body to which the ephemeris applies, relative to the center specified by the segment's descriptor. The epochs corresponding to the states are barycentric dynamical times (TDB), expressed as seconds past J2000.

Type 18 SPK segments have the structure shown below:

+-----------------------+
| Packet 1              |
+-----------------------+
| Packet 2              |
+-----------------------+
            .
            .
            .
+-----------------------+
| Packet N              |
+-----------------------+
| Epoch 1               |
+-----------------------+
| Epoch 2               |
+-----------------------+
            .
            .
            .
+-----------------------+
| Epoch N               |
+-----------------------+
| Epoch 100             | (First directory)
+-----------------------+
            .
            .
            .
+-----------------------+
| Epoch ((N-1)/100)*100 | (Last directory)
+-----------------------+
| Subtype code          |
+-----------------------+
| Window size           |
+-----------------------+
| Number of packets     |
+-----------------------+

In the diagram, each box representing a packet corresponds to either twelve or six double precision numbers; the other boxes represent individual double precision numbers. The number of states normally should be greater than or equal to the window size, which is related to the polynomial degree as shown:

Subtype 0:     DEGREE  =  2 * WINDOW_SIZE  -  1
Subtype 1:     DEGREE  =      WINDOW_SIZE  -  1

The set of time tags is augmented by a series of directory entries; these entries allow the type 18 reader to search for states more efficiently. The directory entries contain time tags whose indices are multiples of 100. The set of indices of time tags stored in the directories ranges from 100 to

(  (N-1) / 100  ) * 100

where N is the total number of time tags. Note that if N is

Q * 100

then only

Q - 1

directory entries are stored, and in particular, if there are only 100 states in the segment, there are no directories. To facilitate the creation of type 18 segments, a segment writing routine called spkw18() has been provided. This routine takes as input arguments the handle of an SPK file that is open for writing, the information needed to construct the segment descriptor, and the data to be stored in the segment. The header of the subroutine provides a complete description of the input arguments and an example of its usage.

Type 19: ESOC/DDID Piecewise Interpolation

As with SPK type 18, SPK type 19 has been provided to support accurate duplication by the SPK system of spacecraft ephemerides used by the European Space Agency (ESA) on the Mars Express, Rosetta, SMART-1 and Venus Express missions.

SPK type 19 is an enhanced version of SPK type 18. Type 19 enables creation of SPK files representing the same ephemerides that can be represented using type 18, but containing far fewer segments. Data from multiple type 18 segments can be stored in a single type 19 segment, as long as those segments satisfy certain restrictions:

• The segments are for the same body, center, and reference frame.

• The segments' coverage intervals, when arranged in increasing time order, overlap only at their endpoints, and have no intervening gaps.

Within a type 19 segment, each set of data corresponding to a type 18 segment is called a mini-segment. A type 19 segment contains one or more mini-segments. Each mini-segment contains a time ordered, strictly increasing sequence of epochs (no two epochs of the same mini-segment may coincide) and an associated sequence of ephemeris data sets called packets. The composition of a packet depends on the subtype of the mini-segment to which the packet belongs; subtypes are discussed in more detail below.

The time coverage of a mini-segment is called an interpolation interval. The endpoints (boundaries) of each interpolation interval must be contained in the time interval bounded by the first and last members of the epoch sequence of the corresponding mini-segment. If the Ith mini-segment's epoch sequence is

E_I1, ..., E_IM

and the mini-segment's interpolation interval bounds are

IV_IB, IV_IE

then it is required that

E_I1 < IV_IB < IV_IE < E_IM
     -              -

Mini-segments are allowed to contain padding epochs and packets beyond both ends of their interpolation intervals. Padding epochs on the left of an interpolation interval are less than the interval start time; padding epochs on the right exceed the interval stop time. Padding enables control of interpolation behavior at and near interpolation interval boundaries. Padding does not contribute to a type 19 segment's time coverage. The use of padding is discussed in greater detail below. The interpolation intervals of a type 19 segment have no intervening gaps and overlap only at single points. The end time of each interpolation interval is the start time of the next. The start time of a type 19 segment is greater than or equal to the start time of the first interval, and the segment's end time is less than or equal to the stop time of the last interval.

Interpolation intervals must have strictly positive length.

When type 19 data are interpolated to produce a state vector for a given request time, only data from a single mini-segment whose interpolation interval contains the request time are used.

When a request time coincides with the boundary between two interpolation intervals, there is a choice as to which interval will provide ephemeris data. The creator of a type 19 segment can control this behavior via a parameter passed to the type 19 segment writer spkw19_c. For a given type 19 segment, depending on the value of this parameter, either the earlier interval is always selected, or the later interval is always selected.

Because of the possibility of evolution of the mathematical representations of ephemerides used by ESA, SPK type 19 is designed to accommodate multiple representations of state data, thereby avoiding a proliferation of SPK data types. SPK type 19 refers to each supported mathematical representation of ephemeris data as a subtype.

Currently SPK type 19 supports three subtypes:

1. Subtype 0

• Separate sliding-window Hermite interpolation of position and velocity. The ephemeris is represented by a series of 12-element packets and associated time tags. The time tags may be unequally spaced. Each packet contains three Cartesian position components, three velocity components meant to be used for Hermite interpolation of the position, three velocity components (not necessarily equal to the previous three), and three acceleration components meant to be used with the second set of velocity components for Hermite interpolation of the velocity. The position and velocity resulting from this interpolation method are in principle independent. The same interpolation degree is used for each position and velocity component.

• The interpolation degree of a subtype 0 mini-segment must be equivalent to 3 mod 4, that is, it must be in the set

{ 3, 7, 11, ..., MAXDEG }

• where MAXDEG is the maximum supported degree.

1. Subtype 1

• Separate sliding-window Lagrange interpolation of position and velocity. The ephemeris is represented by a series of 6-element packets and associated time tags. The time tags may be unequally spaced. Each packet contains three Cartesian position components and three velocity components. The position components and velocity components are interpolated separately. The position and velocity resulting from this interpolation method are in principle independent. The same interpolation degree is used for each position and velocity component.

• The interpolation degree of a subtype 1 mini-segment must be odd and must be in the range 1:MAXDEG, where MAXDEG is the maximum supported degree.

1. Subtype 2

• Sliding-window Hermite interpolation of position and velocity. The ephemeris is represented by a series of 6-element packets and associated time tags. The time tags may be unequally spaced. Each packet contains three Cartesian position components and three velocity components. The position components and velocity components are interpolated together.

• The interpolation degree of a subtype 2 mini-segment must be equivalent to 3 mod 4, that is, it must be in the set

{ 3, 7, 11, ..., MAXDEG }

• where MAXDEG is the maximum supported degree.

The sliding-window interpolation technique used by this data type works as follows: for any request epoch, the data type's state evaluation code computes a component of position or velocity by interpolating a set of values of that component defined on a set of consecutive time tags---a "window"---centered as closely as possible to the request epoch. The nominal window size is dictated by the degree and type (Hermite vs. Lagrange) of the interpolating polynomials. Normally the window of time tags has even size, and the window is selected so that the request time is located between the two central time tags in the window. When the request time is too close to an endpoint of the mini-segment's epoch sequence to permit construction of a window of nominal size, the window is truncated at that endpoint. Although type 19 interpolation intervals support padding, padding is not required. Below we'll discuss the role of padding, but the reader should keep in mind that the size of the pads at either end of an interpolation interval could be zero.

In SPK type 19, interpolation interval padding boundaries (the start time of the padding preceding the interval's coverage and the stop time of the padding following the coverage) affect interpolation in the same way that segment boundaries affect type 18 interpolation. When the request time is near a padding boundary, the window is truncated if necessary on the side closest to the boundary. If an interpolation interval, including padding, contains too few packets to form a window of nominal size, as many packets as are needed and available are used to construct the window. In this case the window size may be odd. In any case the window never includes more than WNDSIZ/2 time tags on either side of the request time, where WNDSIZ is the nominal window size.

The mini-segments of a type 19 segment need not use the same subtypes and interpolation degrees.

The states in a type 19 segment are geometric: they do not take into account aberration corrections. The position and velocity components of each packet represent the position (x, y, z, in kilometers and kilometers per second) of the body to which the ephemeris applies, relative to the center specified by the segment's descriptor. The epochs corresponding to the states are barycentric dynamical times (TDB), expressed as seconds past J2000.

Type 19 SPK segments have the structure shown below:

+--------------------------------+
| Interval 1 mini-segment        |
+--------------------------------+
                .
                .
                .
+--------------------------------+
| Interval N mini-segment        |
+--------------------------------+
| Interval 1 start time          |
+--------------------------------+
                .
                .
                .
+--------------------------------+
| Interval N start time          |
+--------------------------------+
| Interval N stop time           |
+--------------------------------+
| Interval start 100             | (First interval directory)
+--------------------------------+
                .
                .
                .
+--------------------------------+
| Interval start (N/100)*100     | (Last interval directory)
+--------------------------------+
| Mini-segment 1 start pointer   |
+--------------------------------+
                .
                .
                .
+--------------------------------+
| Mini-segment N start pointer   |
+--------------------------------+
| Mini-segment N stop pointer    |
+--------------------------------+
| Boundary choice flag           |
+--------------------------------+
| Number of intervals            |
+--------------------------------+

Below we first describe the overall segment structure, then we cover the mini-segment structure. The array of interval boundaries contains the start time of each interval, plus the stop time of the final interval.

The list of interpolation interval boundary times has its own directory, which has the same structure as the time tag directories of type 18 segments. Let the interval count be N. As with time tag directories, the start time directory contains boundary times whose indices are multiples of 100, except that if N+1 is a multiple of 100, the last boundary time is not included.

The array of mini-segment pointers contains a pointer to the start of each mini-segment, plus a final stop pointer for the final mini-segment. The stop pointer points to the location immediately following the last address of the final mini-segment.

The mini-segment pointers are 1-based indices relative to the start address of the segment. For example, a pointer value of 1 indicates the first address of the segment.

Following the mini-segment pointers is the interval selection flag. When this flag has the value 1.D0, the later interpolation interval is used when a request time falls on the common boundary between two interpolation intervals. If the selection flag is 0, the earlier interval is used.

Each mini-segment has the structure of a type 18 SPK segment. The structure is shown below:

+-----------------------+
| Packet 1              |
+-----------------------+
            .
            .
            .
+-----------------------+
| Packet M              |
+-----------------------+
| Epoch 1               |
+-----------------------+
            .
            .
            .
+-----------------------+
| Epoch M               |
+-----------------------+
| Epoch 100             | (First time tag directory)
+-----------------------+
            .
            .
            .
+-----------------------+
| Epoch ((M-1)/100)*100 | (Last time tag directory)
+-----------------------+
| Subtype code          |
+-----------------------+
| Window size           |
+-----------------------+
| Number of packets     |
+-----------------------+

In the mini-segment diagram, each box representing a packet corresponds to either twelve or six double precision numbers; the other boxes represent individual double precision numbers. The number of packets normally should be greater than or equal to the window size, which is related to the polynomial degree as shown:

Subtype 0:     DEGREE  =  2 * WINDOW_SIZE  -  1
Subtype 1:     DEGREE  =      WINDOW_SIZE  -  1
Subtype 2:     DEGREE  =  2 * WINDOW_SIZE  -  1

The mini-segment's set of time tags is augmented by a series of directory entries; these entries allow the type 19 reader to search for packets more efficiently. The directory entries contain time tags whose indices are multiples of 100. The set of indices of time tags stored in the directories ranges from 100 to

(  (M-1) / 100  ) * 100

where M is the total number of time tags. Note that if M is

Q * 100

then only

Q - 1

directory entries are stored, and in particular, if there are only 100 states in the segment, there are no directories. Following the time tag directory are three parameters associated with the mini-segment: the subtype, the interpolation window size, and the packet count.

To facilitate the creation of type 19 segments, a segment writing routine called spkw19_c has been provided. This routine takes as input arguments the handle of an SPK file that is open for writing, the information needed to construct the segment descriptor, and the data to be stored in the segment. The header of the subroutine provides a complete description of the input arguments and an example of its usage.

Type 20: Chebyshev (velocity only)

SPK data type 20 contains Chebyshev polynomial coefficients for the velocity of a body, relative to its center of motion, as a function of time. The position of the body is obtained by integrating the velocity using a specified integration constant.

This data type is provided to accurately represent EPM ephemerides developed by the Institute of Applied Astronomy (IAA), Russian Academy of Sciences (RAS).

Each type 20 segment contains an arbitrary number of logical records. Each record contains a set of Chebyshev coefficients valid throughout an interval of fixed length. Each record also contains a position vector applicable at the midpoint of its coverage interval.

The records within a segment are ordered by increasing initial epoch. All records contain the same number of coefficients. A segment of this type is structured as

+---------------+
| Record 1      |
+---------------+
| Record 2      |
+---------------+
  .
  .
  .
+---------------+
| Record N      |
+---------------+
| DSCALE        |
+---------------+
| TSCALE        |
+---------------+
| INITJD        |
+---------------+
| INITFR        |
+---------------+
| INTLEN        |
+---------------+
| RSIZE         |
+---------------+
| N             |
+---------------+

A set of seven parameters at the end of the segment provides the information needed to determine the location of the record corresponding to a particular epoch and to determine the units associated with the data:

1. DSCALE is the distance scale used for both position and velocity; DSCALE has units of km. For example, if the distance units are AU, then DSCALE is the value of the AU in km.

2. TSCALE is the time scale used for velocity; TSCALE has units of TDB seconds. For example, if the time units of the velocity data are TDB Julian days, then TSCALE is 86400.

3. INITJD is the integer part of the TDB Julian date of the initial epoch of the first record. INITJD has units of Julian days. INITJD may be less than, equal to, or greater than the initial epoch.

4. INITFR is the fractional part of the TDB Julian date of the initial epoch of the first record. INITFR has units of Julian days. INITFR has magnitude strictly less than 1 day. The sum INITJD + INITFR equals the TDB Julian date of the initial epoch of the first record.

5. INTLEN is the length of the interval covered by each record, in TDB Julian days.

6. RSIZE is the total size of (number of array elements in) each record. The same number of coefficients is always used for each component, and all records are the same size. RSIZE is 3 + 3*(DEGP+1), where DEGP is the common degree of the Chebyshev expansions for each velocity component.

7. N is the number of records contained in the segment.

Each record is structured as follows:

+------------------+
| X  data          |
+------------------+
| Y  data          |
+------------------+
| Z  data          |
+------------------+

where each data section for coordinate I contains

+-------------------------------------------------+
| Chebyshev coefficients for velocity component I |
+-------------------------------------------------+
| Position component I at interval midpoint       |
+-------------------------------------------------+

The velocity coefficients have units of DSCALE km/TSCALE seconds: multiplying a Chebyshev expansion's value by DSCALE/TSCALE converts velocity to units of km/s. The position at a record's midpoint epoch is given in units of DSCALE km: multiplying the position by DSCALE converts the position to units of km.

Type 20 data are used to compute states as follows: for a given time T seconds past J2000 TDB, let MID and RADIUS be the midpoint and radius, expressed as seconds past J2000 TDB, of the record coverage interval that contains T: the coverage interval is the time span

MID - RADIUS : MID + RADIUS

The velocity at T of the body relative to its center of motion is given by the value of the corresponding record's Chebyshev expansions at S, where

S = (T - MID) / RADIUS

The position of the body relative to its center of motion at T is given by

                                      S
(Position at MID) +  RADIUS*( Integral ( Velocity ) )
                                      0

The function spke20_ contains the algorithm used to construct a state from a particular logical record. To facilitate the creation of Type 20 segments, a segment writing function called spkw20() has been provided. This function takes as input arguments the handle of an SPK file that is open for writing, the information needed to construct the segment descriptor, and the data to be stored in the segment. The header of the function provides a complete description of the input arguments and an example of its usage.

Type 21: Extended Modified Difference Arrays

SPK data type 21 contains extended Modified Difference Arrays (MDA), also called difference lines. These data structures use the same mathematical trajectory representation as SPK data type 1, but type 21 allows use of larger, higher-degree MDAs.

This data type is normally used for spacecraft whose ephemerides are produced by JPL's principal trajectory integrator---DPTRAJ. Difference lines are extracted from spacecraft trajectory files (P-files and PV-files ) created by DPTRAJ.

Each segment containing Modified Difference Arrays contains an arbitrary number of logical records. Each record contains difference line coefficients applicable over a time interval containing a reference epoch, along with the state at that epoch. The time intervals of adjacent records overlap at their common endpoints.

The contents of the records themselves are described in [163]. The function spke21_ contains the algorithm used to construct a state from a particular record and epoch.

The records within a segment are ordered by increasing final epoch. The final epochs associated with the records must be distinct.

A segment of this type is structured as follows:

+-----------------------------------------+
| Record 1 (difference line coefficients) |
+-----------------------------------------+
| Record 2 (difference line coefficients) |
+-----------------------------------------+
  .
  .
  .
+-----------------------------------------+
| Record N (difference line coefficients) |
+-----------------------------------------+
| Epoch 1                      |
+------------------------------+
| Epoch 2                      |
+------------------------------+
  .
  .
  .
+------------------------------+
| Epoch N                      |
+------------------------------+
| Epoch 100                    |   (First directory epoch)
+------------------------------+
| Epoch 200                    |   (Second directory epoch)
+------------------------------+
  .
  .
  .
+------------------------------+
| Epoch (N/100)*100            |   (Final directory epoch)
+------------------------------+
| Difference line size         |
+------------------------------+
| N                            |
+------------------------------+

The number of records in a segment, N, can be arbitrarily large. Records 1 through N contain the difference line coefficients and other constants needed to compute state data. Each one of these records contains DLSIZE double precision numbers, where DLSIZE is in the range

..code-block:

71 : (4*MAXTRM) + 1

inclusive. MAXTRM is declared in the SPICELIB include file spk21.inc. A list of the final epochs of the records is stored immediately after the last record.

Following the list of epochs is a second list, the directory containing every 100th epoch from the previous list. If there are N epochs, there will be N/100 directory epochs. If there are fewer than 100 epochs, then the segment will not contain any directory epochs. Directory epochs are used to speed up access to desired records.

The penultimate element of the segment is the difference line size. The final element in the segment is the number of records contained in the segment, N.

The index of the record providing ephemeris data for a user-specified epoch is the index of the first epoch in the segment's epoch list not less than the specified epoch.

Appendix A --- Summary of SP-kernel Functions

Summary of Mnemonics

CSPICE contains a family of functions that are designed specifically for use with SPK files. The name of each function begins with the letters spk, followed by a two- or three-character mnemonic. For example, the function that returns the state of one body with respect to another is named spkez(), pronounced S-P-K-E-Z .

Many of the lower-level CSPICE functions have SPICELIB counterparts implemented in Fortran as entry points of another function.

The following is a complete list of mnemonics and translations, in alphabetical order.

Implemented CSPICE wrappers:

• furnsh()

• spk14a()

• spk14b()

• spk14e()

• spkacs()

• spkapo()

• spkaps()

• spkcls()

• spkcov()

• spkcpo()

• spkcpt()

• spkcvo()

• spkcvt()

• spkez()

• spkezp()

• spkezr()

• spkgeo()

• spkgps()

• spklef()

• spkltc()

• spkobj()

• spkopa()

• spkopn()

• spkpds()

• spkpos()

• spkpvn()

• spksfs()

• spkssb()

• spksub()

• spkuds()

• spkuef()

• spkw02()

• spkw03()

• spkw05()

• spkw08()

• spkw09()

• spkw10()

• spkw12()

• spkw13()

• spkw15()

• spkw17()

• spkw18()

• spkw20()

• unload()

Summary of Calling Sequences

The calling sequences for the SPK functions are summarized below. The functions are grouped by purpose.

High level routines for loading, unloading files:

• furnsh()

• unload()

Lower level routines for loading, unloading files:

• spklef()

• spkuef()

Getting coverage summary:

• spkobj()

• spkcov()

Computing states and positions:

• spkezr()

• spkpos()

• spkez()

• spkezp()

• spkapo()

• spkpvn()

• spkssb()

• spkgeo()

• spkgps()

Low-level routines for computing states and positions:

• spkacs()

• spkaps()

• spkltc()

Computing states using constant-velocity or constant-position objects:

• spkcpo()

• spkcpt()

• spkcvo()

• spkcvt()

Selecting files, segments:

• spksfs()

Writing segments to files:

• spkpds()

• spkw02()

• spkw03()

• spkw05()

• spkw08()

• spkw09()

• spkw10()

• spkw12()

• spkw13()

• spkw15()

• spkw17()

• spkw18()

• spkw20()

Examining segment descriptors:

• spkuds()

Extracting subsets of data from a segment:

• spksub()

To write new or append segments to SPK files:

• spkopn()

• spkopa()

• spkcls()

Appendix B --- A Template for SPK Comments

An undocumented ephemeris is in many respects worse than undocumented source code. With source code you can at least read the code and perhaps discern the function of the source code. An ephemeris on the other hand is a binary file. All it contains are numbers. It's very difficult to determine the purpose of an ephemeris simply from the state information it contains. For this reason, any ephemeris created for use by anyone other than yourself needs documentation.

If you create SPK files NAIF strongly recommends that you include descriptive documentation in the comments portion of the SPK file. You can use the utility program COMMNT to insert comments into the file, or you may use the functions in the SPC family to insert the comments when you create the SPK file. (See commnt.ug or spc.req for further details.)

This appendix addresses the contents of your comments. What will others (or yourself) want to know about the SPK file weeks, months or years after it has been created? Providing this information can be a challenge. It's difficult to know in advance all the questions someone might ask about an ephemeris you've created. To assist with this task NAIF has devised a template that you may wish to use as a starting point when creating the comments for an SPK file.

Constraints

The comments you place in an SPK file must be plain ASCII text. Each line of text must consist of 80 or fewer characters. The text must contain only printing characters (ASCII characters 32 through 126).

The Basic Template

Here's one way to create the comments for an SPK file.

Objects in the Ephemeris

List the names and NAIF ID codes for the objects in the file.

Approximate Time Coverage

Provide a summary of the time for which states are available for the objects in the file. If you use UTC times in this summary and the ephemeris extends more than 6 months into the future, you should probably state that the times are approximate. You don't know when leapseconds will occur more than a few months in advance, so you can't know the exact UTC time boundaries for the ephemeris if it extends years into the future.

Status

Provide the status of the ephemeris. Tell the user why this ephemeris was created and for whom it is intended. For example, if this is the second in a series of ephemerides that will be produced for some object tell which ephemeris this one supersedes. Tell the user when the next ephemeris in the series will be available. Is the ephemeris suitable only for preliminary studies? Is it good for all Earth based observations? Is this an official operational product? Are there situations for which the ephemeris is not suitable?

Pedigree

Provide a production summary for the ephemeris. Tell when the ephemeris was produced (the system time stamp may not port if the file is copied to other systems). Say who produced the ephemeris; what source products were used in the production; what version of the producing program was used in the creation of the ephemeris. If the ephemeris is based on a set of recent observations, say so. In short give the user the pedigree of this ephemeris. This information is mostly for your benefit. If a problem arises with the ephemeris, you will know how the problem was created and have a better chance of fixing the problem.

Usage

Provide information the user will need to effectively use the ephemeris. Tell the user what other SPICE kernels are needed to use this ephemeris. For example, if the ephemeris contains only the state of an asteroid relative to the sun, the user will probably need a planetary ephemeris to effectively use the one you've created. Recommend a planetary ephemeris to use with your SPK file. If the ephemeris contains states of objects relative to non-inertial frames, the user will probably need other kernels so that various state transformations can be performed. Recommend which of these kernels the user should use with your SPK file.

Accuracy

If possible give some estimate as to the accuracy of your SPK file. Use numbers. Words such as this is the best available do not convey how much you know about the ephemeris.

Special Notes

Provide a description of any special properties of this ephemeris. For example, if some observation seems to be in conflict with this ephemeris you should probably point this out.

References

List any references that may be relevant to the understanding of the ephemeris. For example, if the ephemeris is based upon observations contained in the literature, site the appropriate articles. If there is some technical memorandum or private communication that addresses certain aspects of this ephemeris list it. This will allow you to more easily answer questions about the ephemeris.