NASA Technical Memorandum 104566, Vol. 11

SeaWiFS Technical Report Series

Stanford B. Hooker, Elaine R. Firestone,

and A.W. Indest, Editors

Volume 11, Analysis of Selected Orbit Propagation

Models for the SeaWiFS Mission

Frederick S. Patt, Charles M. Hoisington, Watson W. Gregg

and Patrick L. Coronado

June 1993
NASA Technical Memorandum 104566, Vol. 11

SeaWiFS Technical Report Series

Stanford B. Hooker, Editor

NASA Goddard Space Flight Center

Greenbelt, Maryland

Elaine R. Firestone and A.W. Indest, Technical Editors

General Sciences Corporation

Laurel, Maryland

Volume 11, Analysis of Selected Orbit Propagation

Models for the SeaWiFS Mission

Frederick S. Patt

General Sciences Corporation

Laurel, Maryland

Charles M. Hoisington

Science Systems Applications, Inc.

Lanham, Maryland

Watson W. Gregg

Patrick L. Coronado

NASA Goddard Space Flight Center

Greenbelt, Maryland

National Aeronautics and

Space Administration

Goddard Space Flight Center

Greenbelt, Maryland 20771

1993
F.S. Patt, C.M. Hoisington, W.W. Gregg, and P.L. Coronado

Abstract

An analysis of orbit propagation models was performed by the Mission Operations element of the Sea-viewing

Wide Field-of-view Sensor (SeaWiFS) Project, which has overall responsibility for the instrument scheduling.

The orbit propagators selected for this analysis are widely available general perturbations models. The analysis

includes both absolute accuracy determination and comparisons of different versions of the models. The results

show that all of the models tested meet accuracy requirements for scheduling and data acquisition purposes. For

internal Project use the SGP4 propagator, developed by the North American Air Defense (NORAD) Command,

has been selected. This model includes atmospheric drag effects and, therefore, provides better accuracy. For

High Resolution Picture Transmission (HRPT) ground stations, which have less stringent accuracy requirements,

the publicly available Brouwer-Lyddane models are recommended. The SeaWiFS Project will make available

portable source code for a version of this model developed by the Data Capture Facility (DCF).

1. INTRODUCTION

The Sea-viewing Wide Field-of-view Sensor (SeaWiFS)

Project is a Code 970.2 activity at the National Aeronautics and Space Administration (NASA) Goddard Space

Flight Center (GSFC). The primary responsibility of the

Mission Operations element is to provide command schedules to maximize performance. Producing these command

schedules requires the propagation of orbit positions over

periods of days to weeks. It is of central importance to

the successful performance of Mission Operations to select

and use an orbit propagation model that will be accurate

over these time scales.

A secondary responsibility of the Mission Operations

element is to provide pointing vectors and orbit models

to worldwide ground stations to enable the acquisition of

SeaWiFS data, which are directly broadcast in real time.

These direct broadcast data are in High Resolution Picture Transmission (HRPT) format. The time scales for

these activities are typically much less than for command

planning (usually less than three days).

The purpose of this paper is to assess several widely

available orbit propagation models for use by Mission Operations. Selection of orbit models for use by Mission Operations follows three steps: 1) collect requirements for

performance, 2) assess availability of candidate models,

and 3) analyze performance in relation to requirements.

This analysis was a joint effort by the Mission Operations

element and the SeaWiFS Data Capture Facility (DCF),

which is responsible for supporting the HRPT ground stations and has also implemented some of the tested orbit

prediction models.

2. BACKGROUND

SeaWiFS is designed to make routine, global observations of ocean color for a five-year mission lifetime. It is

a follow-on sensor to the highly successful Coastal Zone

Color Scanner (CZCS), which was carried on NIMBUS-7

in operation from 1978 to 1986. The SeaWiFS instrument

will be contained on the SeaStar spacecraft, on which it is

the sole occupant.

In a unique arrangement, the sensor, spacecraft, and

launch vehicle (Pegasus) are being built and will be operated by Orbital Sciences Corporation (OSC) of Chantilly,

Virginia. NASA’s role will be to purchase the SeaWiFS

data from OSC and provide command schedules to enable

global coverage of the Earth and facilitate calibration of

the sensor. Mission Operations serves as the component

which determines these schedules and passes them to OSC

for uploading.

SeaStar will be placed in a 705 km, sun-synchronous,

near-local-noon descending node orbit. These and other

orbit characteristics are summarized in Table 1.

Table 1. SeaStar spacecraft orbit parameters.

Orbit Parameter Value

Altitude 705 km

Eccentricity < 0.002

Orbital Repeat Time 16 days (233 orbits)

Period 98.9 minutes

Inclination 98.2◦

Equator Crossing Time Noon Local Time

Node Type Descending

Successive Equatorial

Crossing Longitude −24.721◦

2.1 Mission Operations

As stated earlier, the primary role of the Mission Operations element is to provide OSC with schedules of command sequences to maximize the coverage and scientific

usefulness of SeaWiFS data and ensure data acquisition.

All of these commands are scheduled according to viewing conditions or events which are a direct function of the

spacecraft orbit position. The commands under the responsibility of Mission Operations include: sensor electron1
Analysis of Selected Orbit Propagation Models for the SeaWiFS Mission

ics on and off times, data down-link times at NASA’s Wallops Flight Facility (WFF), tilt change times, gain changes,

calibration times, and broadcast times for research HRPT

stations.

2.2 Orbit Propagation Models

There are basically two classes of orbit propagation

models: general perturbations models (GPMs) and special

perturbations models (SPMs). GPMs are reformulations

of the equations of motion so that an analytical solution

may be found. Typically, they include only the perturbing

forces of a low order Earth gravity field, although, some

also may contain atmospheric drag. A common example is

the Brouwer-Lyddane model. GPMs are fast and computationally inexpensive at the cost of a reduction in accuracy.

SPMs are characterized by the requirement of numerical integration of the equations of motion. Such methods

are computationally expensive and output is much slower

than for GPMs. The advantage is that they can take

into account more perturbing forces, such as a high-order

Earth gravity field, multiple body gravitational forces, atmospheric drag, solar radiation, tides, and others. Consequently, SPMs can be very accurate, but they can also

be more unstable and are intimately sensitive to the initial

conditions.

2.3 Orbital Elements

A spacecraft orbit can be parameterized in several ways.

Three common representations are: mean element sets, osculating elements, and orbit state vectors. Both mean and

osculating element sets include the six classical Keplerian

elements (semi-major axis, eccentricity, inclination, right

ascension of the ascending node, argument of perigee and

mean anomaly), but in fact they have quite different interpretations. Mean element sets represent an average of

the orbit and are specifically designed for use with GPMs.

Commonly distributed orbital elements sets by the U.S.

Space Command (formerly the North American Air Defense, or NORAD), Naval Space Surveillance (NAVSPASUR), and the National Oceanic and Atmospheric Administration (NOAA) are all mean elements. Mean element

sets are not used directly to compute the spacecraft position, but are converted to osculating elements for this

purpose.

Osculating elements are an instantaneous Keplerian

representation of the orbit; the osculating elements vary

significantly over the orbit for low Earth spacecraft and

are not readily propagated. Orbit state vectors are defined

as the Cartesian position and velocity vectors in the geocentric reference frame. Orbit state vectors are frequently

used as the input to SPMs and are the standard output for

all models. Orbit state vectors, or derived quantities such

as latitude, longitude, and altitude, are the information

required for scheduling.

The SeaWiFS Project has two potential sources of orbital parameters: distributed element sets such as those

from NORAD or NAVSPASUR, or the Global Positioning

System (GPS) data included in the spacecraft down link.

While the GPS data is considered the primary source of

orbit position information for navigation, at present there

is no proven method for using this data in an orbit propagation model. Thus, distributed element sets are to be

considered the first choice of orbit parameters.

2.4 Requirements Analysis

The most critical requirement for orbit propagation accuracy is to provide antenna pointing information for the

down-link times at WFF. The Wallops antenna to be used

for acquisition of SeaWiFS data is 9 m in diameter and

autotracking with x-y tracking capability. The requirements for acquisition of data by this antenna are 3 seconds

along track, and 0.5◦ in azimuth, for a two-day propagation. A lesser requirement is to provide predicted downlink times three weeks in advance for conflict analysis, for

which the accuracy required is two minutes along track.

The requirement for acquisition sets the limit for accuracy

for the mission; none of the other commands have such a

stringent accuracy requirement.

Regarding HRPT ground stations, the SeaStar satellite

L-band down-link has been designed to resemble all of the

transmission parameters of the TIROS satellites. This was

done in order to minimize the impact on small satellite

ground stations around the world. The SeaWiFS Project,

in turn, has adopted the TIROS data quality requirements,

with a maximim bit error rate (BER) of 10−6. In order for

this BER to be met, OSC has suggested the system gain

to total system noise temperature (G/T), a measure of a

ground station’s quality performance, should be not less

than 6.0.

The antenna tracking requirements are given next. The

L-band tranmission parameters are given in Table 2.

Table 2. SeaStar L-band transmission parameters.

Parameter Value

Frequency 1207.5 MHz ±34.05 KHz

Polarization Right-Hand Circular

Bandwidth 1.2 MHz (−3 db)

Data Rate 665.4 Kbps

The pointing accuracies required for various recommended

antenna dish sizes (including 1◦ of buffer) are shown in

Table 3.

Table 3. Pointing accuracies for different dish sizes.

Dish Size [ft] Pointing Accuracy [ ◦]

8 3.31

6 4.75

5 5.88

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F.S. Patt, C.M. Hoisington, W.W. Gregg, and P.L. Coronado

The SeaWiFS Project does not recommend using an antenna dish size of less than 5 ft since the BER of 10−6

is not guaranteed unless the system noise temperature is

extremely low.

These requirements are interpreted in terms of orbit

prediction accuracies as follows.

1. The 3-second timing requirement, for a typical orbit velocity of 7.5 km s−1, corresponds to

22.5 km along track.

2. The 0.5◦ azimuth error requirement at acquision can also be evaluated in terms of alongtrack errors. At the planned SeaStar altitude,

the station-to-satellite distance is approximately

2,500 km at a typical acquisition of 5◦ above the

horizon. 0.5◦ of azimuth error corresponds to

21.8 km of error in the predicted satellite position perpendicular to the line of sight.

3. For the HRPT stations, the 8-foot dish has the

most stringent requirement at 3.31◦; assuming

that small stations do not have autotracking antennas, this requirement would have to be met

by the orbit propagator throughout the contact.

The orbit propagation errors would have maximum effect when the satellite is directly overhead; at an altitude of 705 km, 3.31◦ corresponds

to 40.7 km.

Thus, the WFF acquisition timing and azimuth requirements are most stringent and roughly equal in terms of

orbit propagation accuracy.

2.5 Approach

The approach presented here is limited to assessing the

feasibility of several widely available GPMs. The fast, computationally inexpensive analytical solutions of such models, in conjunction with mean element sets, are appealing

if the requirements can be met.

3. METHODS

3.1 Selection of Models for Evaluation

The selection of orbit propagation models for evaluation was based on the following criteria:

a) Availability of source code; this was necessary

to allow the model to be configured as either a

standalone program or a subroutine, to enable

it to be run on any of several platforms and also

to allow the format of the output to be closely

tailored to the needs of Mission Operations;

b) Use of known methods and models, with references to published derivations or specification;

and

c) Compatibility with readily available orbital element sets.

For the last criterion, either NORAD two-line element

sets or Navy elements from NAVSPASUR were considered

as readily available for SeaWiFS. The so-called TBUS elements provided by NOAA for their satellites were not

considered, because NOAA is not expected to provide this

service for SeaWiFS.

Two orbit propagation models were selected for evaluation; each was available in two separate implementations. The first was a standard Brouwer-Lyddane model,

an analytic propagator which does not include atmospheric

drag. The two implementations of this were as follows:

a set of routines previously developed by one of the authors (Hoisington) for the DCF, using the original published work of Brouwer (1959) and Lyddane (1963), and

also specifications for the Goddard Trajectory Determination System (GTDS) (Cappellari et al. 1976); and the

BRWLYD routine developed by NOAA (Kidwell 1991).

The latter model is presumably already widely used in the

NOAA and Advanced Very High Resolution Radiometer

(AVHRR) user communities. Brouwer-Lyddane propagators use a mean element set as input; the derivation of this

set from NORAD two-line elements is discussed below.

The second model was the SGP4 low Earth orbit model

developed by NORAD Project Spacetrack. This is a modified Brouwer propagator, which includes an atmospheric

drag term, and is specifically designed to be compatible

with NORAD element sets. This model was also implemented by Hoisington for the DCF from NORAD documentation (Hoots and Roehrich 1980). After the evaluation was started, an updated version of the SGP4 source

code was obtained directly from Project Spacetrack and

was used for additional testing.

As a side note, the SGP4 orbit model has restrictions

placed on its use—it can only be used by U. S. government

agencies and their contractors.

3.2 Orbital Elements Used for Evaluation

The evaluations were performed using NORAD twoline element sets for the following NOAA and Land Resource Satellite (LANDSAT) spacecrafts: NOAA-12, which

is in a polar orbit at a higher altitude than planned for SeaStar (822 versus 705 km); and LANDSAT-4 and 5, which

are in very similar orbits to that planned for SeaStar. Element sets were available at frequent intervals for all three

satellites.

The LANDSAT satellites perform regular orbit adjustment maneuvers, making extended evaluations impossible, since no thrust model is available. NOAA-12 is completely free flying, but at its higher altitude it is expected

to experience a lower atmospheric drag effect. Thus, the

NOAA-12 elements were used for extended studies, while

the LANDSAT elements were used for short-term (up to

10 days) analyses, as possible between orbit maneuvers,

and also to validate the evaluation performed with the

NOAA-12 elements. The actual element sets were obtained

3
Analysis of Selected Orbit Propagation Models for the SeaWiFS Mission

from a bulletin board maintained by GSFC Code 513, and

spanned the dates of July 1 through November 16, 1992.

The use of the NORAD two-line element sets for the

Brouwer-Lyddane models required a conversion since the

NORAD elements are not explicitly mean element sets.

Specifically, the mean motion provided with the NORAD

elements sets must be converted to a semi-major axis. This

conversion does not use the classical form, as described for

Keplerian orbits in standard textbooks. The actual conversion was taken from the SGP4 source code and uses the

mean motion, inclination, and eccentricity (a detailed description is provided in Appendix A). Mean element sets

derived in this manner were found to be entirely satisfactory for use in the Brouwer-Lyddane model.

3.3 Constants Used in Models

All of the models utilize constants to specify the gravitational field terms and Earth radius. In order to compare

the various models and implementations, it was necessary

to choose a consistent set of constants. The constants selected were taken from the system accepted by the International Astrophysical Union (IAU) in 1976 and published

in the 1984 Astronomical Almanac (see Table 4).

Table 4. Modeling constants for the Earth.

Constant Value

Radius, Re 6,378.137 km

Gravitational Constant, Ge 398,600.5 km3 s−2

Gravity Field Terms:

J2 0.0010863

J3 −0.0000254

J4 −0.0000161

The Brouwer-Lyddane models also include a J5 gravity field term, which was not specified in the IAU system

of constants; both of the versions tested had this constant

specified, but with different values. These constants were

retained for the absolute accuracy analyses, but for comparison purposes they were set to zero.

3.4 Evaluation Methods

The evaluation was performed in two parts: 1) determination of the absolute propagation accuracy of each model,

and 2) comparison of the propagations between the models. The former was used to determine whether the models

would meet the accuracy requirements for SeaWiFS Mission Operations, while the latter was performed mainly to

compare different implementations of the same model.

The final output of each model is in the form of Cartesian orbit state vectors (position and velocity) in the geocentric inertial reference frame. The two parts involved

a somewhat different approach to selecting the orbit vectors used for the comparisons, but in each case the actual

vector comparisons were identical.

Determining the absolute accuracy of a propagated orbit requires the availability of a truth model. All of the

orbit information for the subject spacecraft was obtained

from the same source: the NORAD two-line elements sets.

NORAD elements are intended to support scheduling requirements, as described earlier, not high-accuracy applications (such as navigation). In fact they are believed to be

degraded at the level of a few kilometers. This is still sufficient to determine whether the propagated orbit meets the

SeaWiFS scheduling requirements, which are on the order

of tens of kilometers.

The use of NORAD elements at epoch as a truth model

would be ideally validated by comparison with an external source of orbit data. While SeaWiFS Mission Operations does not have access to independent data sources for

the NOAA-12 and LANDSAT orbits used for this analysis,

an evaluation of GPS data from the Extreme Ultraviolet

Explorer (EUVE) satellite is currently being performed.

The NORAD elements for EUVE are also available, but

at longer intervals than for the NOAA-12 and LANDSAT

elements. However, the GPS data, which spans the epoch

time for a few of the NORAD element sets, demonstrates

that the elements are accurate to within one kilometer at

epoch.

It was therefore assumed that the orbit vector determined from each element set at its epoch time would be a

reasonable truth model since no propagation was involved,

just a conversion from mean elements to vectors. Thus,

the approach in determining absolute propagation accuracy was as follows. For each element set, an orbit state

truth vector was computed at the epoch time. Then each

element set was used to propagate the orbit to the epoch

time of every other element set. This generated a number

of orbit vectors representing various propagation times, all

of which were compared with the truth vectors.

The comparisons between models used a simpler approach. For a given element set, orbit vectors were generated at fixed intervals for a specified period using each

model. In this type of evaluation, no particular set of vectors could be designated as truth, since all were generated

by propagation; the point is to compare vectors from different models which have been propagated for the same

time since epoch.

For both types of evaluations, the comparisons were

performed as follows. The difference between two orbit

position vectors was computed by subtracting each Cartesian component (for the absolute accuracy evaluation the

truth vector was subtracted from the propagated vector;

for the comparisons the order of subtraction is purely arbitrary). The difference vectors were then converted to

along track, cross track and radial (i.e., in the direction of

the position vector) components. Determination of these

components was performed by first determining the unit

vectors in the directions of the orbit velocity, orbit normal

(computed as the vector cross product of the position and

velocity vectors), and orbit position. Specifically, for posi4
F.S. Patt, C.M. Hoisington, W.W. Gregg, and P.L. Coronado

tion and velocity vectors P and V , and an orbit position

difference vector D ,

O = P × V ,  (1)

Dat = D · V

|V |

, (2)

Dct = D · O

|O|

, (3)

and

Drad = D · P

|P|

, (4)

where Dat is the along-track position difference, Dct is the

cross-track position difference, and Drad is the radial position difference.

4. RESULTS AND DISCUSSION

The orbit propagation accuracy requirements for SeaWiFS are stated for periods of 7 and 21 days. The results

are shown for periods of 10 and 30 days, to demonstrate

how well the requirements are met beyond the minimum

times. Along-track propagation differences are expressed

in both kilometers and seconds, since the primary accuracy

requirements are stated in terms of timing errors.

4.1 Absolute Accuracy Results

This evaluation used 70 sets of NOAA-12 NORAD elements spanning the period July 1 through November 16,

1992. The plots of absolute along-track propagation errors versus days since epoch for NOAA-12 are shown in

Figs. 1–8. The results produced by SGP4, DCF/SGP4,

DCF/Brouwer, and NOAA/Brouwer are shown for propagations of 10 and 30 days.

The most significant factor in the absolute accuracy of

the orbit propagations is the presence of a drag term in the

SGP4 routines and the lack of such a term in the BrouwerLyddane routines. The drag model allows the mean alongtrack propagation error of the SGP4 routines to be near

zero, even after 30 days, while the Brouwer propagations

always show systematic (negative) errors, with what appears to be a second order time dependence.

All of the models met the stated timing accuracy requirements of 3 seconds after 2 days and 120 seconds after

21 days. In fact, the 3-second requirement for acquisition

was not exceeded for more than 7 days in all cases. The

largest errors observed after 7 days were approximately

10 km, or 1.5 seconds, for both of the SGP4 routines and

15 km, or 2 seconds, for DCF/Brouwer. After 21 days,

the largest errors were less than 15 seconds for all models.

The plots for the SGP4 models showed a dispersion around

roughly zero mean (Figs. 1–4), while the Brouwer models

clearly showed the consistent degradation in accuracy from

the lack of a drag term (Figs. 5–8).

The azimuth requirement, equivalent to 21.8 km along

track after 2 days, was also met by all of the models as

shown by Figs. 1–8. The maximum errors after 7 days

were approximately 10 km for the SGP4 models and 12 km

for the Brouwer models.

The most stringent HRPT station requirement, which

is 40.7 km along track for an 8-foot dish, was easily met by

all models well beyond 7 days. The Brouwer propagation

errors were less than 40 km. for approximately 12 days,

while the SGP4 models meet this requirement for at least

two weeks.

The other components of the propagation errors (cross

track and radial) were consistently much smaller than the

along-track errors, by as much as two orders of magnitude. Sample plots of the cross-track and radial errors for

DCF/SGP4 over 30 days are shown in Figs. 9 and 10. The

maximum cross-track errors at 7 and 21 days were less than

0.5 km and slightly more than 1 km, respectively, while the

radial errors were less than 0.5 km and 1 km, respectively.

Thus, the effects of orbit propagation on timing and antenna pointing can be assumed to be almost exclusively a

result of the along-track errors.

However, even with the SGP4 models, there are substantial variations in the effectiveness of the drag model.

This depends on the accuracy of the drag term included in

the NORAD element sets. (This term is usually referred

to as BSTAR in the documentation and code; no units are

given.) Examination of the BSTAR term for the NOAA-12

element sets showed large variations around the average,

ranging from near zero to approximately double the average value. The performance of the SGP4 propagation correlates very closely with these variations in the BSTAR term;

the points on the SGP4 plots which show the largest negative trend correspond to a BSTAR of approximately zero,

and show essentially the same behavior as the Brouwer

model (see Figs. 2 and 4).

The cause of the large variations in BSTAR, which is

determined by NORAD along with the other elements, is

unknown. In practice, the extreme values of BSTAR can

easily be found by a cursory examination of the elements

and rejected; this was not done for this evaluation to avoid

skewing the results in favor of the SGP4 model. Filtering

of the elements sets in this manner would be expected to

reduce the maximum SGP4 propagation errors to less than

1 second at 7 days and less than 10 seconds at 21 days,

respectively.

As mentioned previously, the NOAA-12 orbit is higher

than the altitude planned for SeaStar (approximately 822

versus 705 km). The atmospheric drag at the higher altitude would be expected to be lower, and therefore the

propagation errors for NOAA-12 may not be considered

truly representative for SeaStar. The LANDSAT-4 and 5

orbits are much closer in altitude to SeaStar; however, due

to the regular performance of orbit maintenance maneuvers, it is not possible to perform 30-day evaluations using

LANDSAT elements. An evaluation was performed using

5
Analysis of Selected Orbit Propagation Models for the SeaWiFS Mission

Fig. 1. SGP4 along-track propagation errors for NOAA-12: 0–10 days.

Fig. 2. SGP4 along-track propagation errors for NOAA-12: 0–30 days.

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F.S. Patt, C.M. Hoisington, W.W. Gregg, and P.L. Coronado

Fig. 3. DCF/SGP4 along-track propagation errors for NOAA-12: 0–10 days.

Fig. 4. DCF/SGP4 along-track propagation errors for NOAA-12: 0–30 days.

7
Analysis of Selected Orbit Propagation Models for the SeaWiFS Mission

Fig. 5. DCF/Brouwer along-track propagation errors for NOAA-12: 0–10 days.

Fig. 6. DCF/Brouwer along-track propagation errors for NOAA-12: 0–30 days.

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F.S. Patt, C.M. Hoisington, W.W. Gregg, and P.L. Coronado

Fig. 7. NOAA/Brouwer along-track propagation errors for NOAA-12: 0–10 days.

Fig. 8. NOAA/Brouwer along-track propagation errors for NOAA-12: 0-30 days.

9
Analysis of Selected Orbit Propagation Models for the SeaWiFS Mission

Fig. 9. DCF/SGP4 cross-track propagation errors for NOAA-12: 0–30 days.

Fig. 10. DCF/SGP4 radial propagation errors for NOAA-12: 0–30 days.

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F.S. Patt, C.M. Hoisington, W.W. Gregg, and P.L. Coronado

Fig. 11. SGP4 along-track propagation errors for LANDSAT-5: 0–10 days.

LANDSAT-5 elements over 10 days to demonstrate the

comparability with NOAA-12, and the results for SGP4,

DCF/SGP4, and DCF/Brouwer are shown in Figs. 11–

13. As these figures show, the 10-day performance for the

SGP4 routines is very similar to that for NOAA-12, while

the DCF/Brouwer errors are slightly larger (approximately

2.5 km after 7 days versus 2 km for NOAA-12). Given that

the NOAA-12 results met the 21-day requirement by a substantial margin, it is expected the SeaWiFS propagations

will meet this requirement as well.

4.2 Comparisons

Comparisons were performed of the different versions of

each of the two models: between the SGP4 and DCF/SGP4

routines, and between the DCF and NOAA versions of

the Brouwer-Lyddane model. Comparisons between SGP4

and Brouwer-Lyddane models were not performed, since

the absolute accuracy evaluations clearly showed significant differences in the outputs of the two models, almost

entirely due to the presence of the drag term in SGP4.

The comparison of the latest SGP4 routine with the

DCF implemented version is of some interest. These routines have a common heritage, i.e., NORAD Project Spacetrack. An examination of the code shows the latest SGP4

implementation has been substantially updated from the

version produced by DCF from the previous Project Spacetrack documentation. The question is whether the changes

were in the logic or involved improvements to the model.

The comparison was performed using a typical set of

NOAA-12 NORAD elements. Orbit vectors were produced

at regular (30-minute) intervals for 30 days. The differences were computed as along-track and cross-track components. The results for the SGP4 models are shown in

Figs. 14 and 15, which indicate that the propagation differences, while not completely negligible, are small compared to the overall propagation errors. The maximum

along-track differences at 7 and 21 days are less than 0.2

second and 0.5 second, respectively, while the maximum

cross-track differences are less than 0.1 km and 0.2 km, respectively. The maximum cross-track differences show very

linear behavior in magnitude with apparently zero mean,

while the along-track differences show some periodic effects with an overall linear trend and a positive mean. The

along-track differences indicate that the mean of the absolute propagation accuracy distribution is closer to zero,

possibly due to better drag modeling in the new SGP4.

The cross-track differences appear to represent a small difference in the precession rate of the orbit plane.

The Brouwer model comparison (Figs. 16 and 17) show

a very small linear trend in the along-track component

(approximately 0.005 second after 21 days). The cross11
Analysis of Selected Orbit Propagation Models for the SeaWiFS Mission

Fig. 12. DCF/SGP4 along-track propagation errors for LANDSAT-5: 0-10 days.

Fig. 13. DCF/Brouwer along-track propagation errors for LANDSAT-5: 0-10 days.

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F.S. Patt, C.M. Hoisington, W.W. Gregg, and P.L. Coronado

Fig. 14. Along-track propagation differences between SGP4 and DCF/SGP4.

Fig. 15. Cross-track propagation differences between SGP4 and DCF/SGP4.

13
Analysis of Selected Orbit Propagation Models for the SeaWiFS Mission

Fig. 16. Along-track propagation differences between DCF/Brouwer and NOAA/Brouwer.

Fig. 17. Cross-track propagation differences between DCF/Brouwer and NOAA/Brouwer.

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F.S. Patt, C.M. Hoisington, W.W. Gregg, and P.L. Coronado

Table A1. The format of the NORAD element set.

A generic NORAD two-line element:

Line 1: sssssU	ly	ld		A	yyddd.ffffffff		mmd1						mmd2						bstar			0		elno

Line 2: sssss		ii.iiii	lll.llll	eeeeeee	www.wwww	mmm.mmmm	nn.nnnnnnnn	rrrrx

Line 1 Variables Line 2 Variables

sssss is the 5-digit spacecraft ID ii.iiii is the inclination (◦)

ly is the spacecraft launch year (not used) lll.llll is the right ascension of the

ld is the spacecraft launch day (not used) ascending node (◦)

yy is the element epoch year (two digits) eeeeeee is the eccentricity (a decimal point

ddd is the element epoch day-of-year is implied at the left)

ffffffff is the element epoch time of day www.wwww is the argument of perigee (◦)

mmd1 is a mean motion time derivative (not used) mmm.mmmm is the mean anomaly (◦)

mmd2 is a mean motion time derivative (not used) nn.nnnnnnnn is the mean motion (revs/day)

bstar is the drag term (mantissa and exponent) rrrr is the orbit number (not used)

elno is the element set number (not used) x is not used

A typical NORAD two-line element set for NOAA-12:

21263U	91	32		A	92183.30748338		.00000257		00000-0		13333-3	0		3228

21263		98.6941	213.0068	0013736	146.1404	214.0628	14.22063660	58773

Note: The bstar term represents a value of 0.00013333 (0.13333E−03) and the eccentricity is 0.0013736. All other values

are formatted explicitly.

track component shows very similar behavior to the SGP4

comparison, with maximum differences of approximately

0.1 km and 0.25 km at 7 days and 21 days, respectively.

This also seems to indicate a small difference in the orbit

plane precession rate. Given the overall performance of

these models, the differences are too small to demonstrate

an advantage for either model.

5. RECOMMENDATIONS

The evaluations have shown that any of the four propagators evaluated, in conjunction with NORAD elements,

are capable of meeting the stated accuracy requirements

for SeaWiFS scheduling purposes. The SGP4 propagators

provide superior results, especially if rudimentary quality

assurance of the element sets is performed. The major

disadvantage of the SGP4 model is the restriction to government agencies and contractors. The recommendations

are, therefore, as follows:

1. The Mission Operations element should use the

SGP4 model with NORAD elements to support

scheduling. The performance of the new SGP4

routine and the DCF/SGP4 version are very

nearly equivalent for this purpose, but the new

SGP4 appears to model drag better.

2. The Brouwer-Lyddane model meets the needs

of all non-government facilities which need SeaWiFS orbit propagations, using NORAD elements converted to mean elements as described

in Appendix A. The requirements for this purpose are less stringent, as stated earlier. The

SeaWiFS Project will provide the DCF BrouwerLyddane propagator to HRPT stations.

Appendix A

The NORAD two-line elements completely specify a spacecraft’s orbit; unfortunately they cannot be used directly with

the Brouwer-Lyddane orbit model. Specifically, the NORAD

model uses the mean motion, whereas the Brouwer model requires the semi-major axis. These two parameters are redundant for classical Keplerian orbits but the conversion is more

complex for non-spherical gravitational fields (i.e., low Earth

orbits) and mean element sets.

The format of the NORAD element set is given in Table A1.

The Brouwer models input the elements as a 6-element array,

where the order of the elements is: semimajor axis (km), eccentricity, inclination, right ascension of the ascending node,

argument of perigee, and mean anomaly. All but the semimajor axis can be copied directly from the NORAD element

set.

The conversion of the mean motion to the semi-major axis

is performed as follows. First, the mean motion is converted

from revolutions per day (n) to radians per minute (xno):

xno = n 2π

1,440 (A1)

The calculation of the semi-major axis uses the gravitational

constant in units of fractional Earth radii (R1.5 e ) per minute,

and also the J2 perturbation term. The Earth radius (Re)

and gravitational constant (Ge) were defined earlier (Table 4)

in units of kilometers and seconds, and J2 (unitless) was also

defined, as follows:

Re = 6,378.137 km (A2)

Ge = 398,600.5 km3 s

−2 (A3)

J2=0.00108263. (A4)

The revised value of the gravitational constant used below

(xke) is:

xke = 60GeR−3 e (A5)

= 0.0743668531 (A6)

15
Analysis of Selected Orbit Propagation Models for the SeaWiFS Mission

where xke is in units of Earth radii1.5 (R1.5 e ) per minute. The

initial (classical) estimate of the semi-major axis (a1) is:

a1 =

xke

xno2/3

(A7)

where a1 is in units of Earth radii.

The perturbation corrections to the semi-major axis use the

inclination (i), the eccentricity (e), and J2 as follows:

temp = 0.75 J2 3 cos2(i) − 1

(1 − e2)1.5 (A8)

del1 = temp

a12 (A9)

a0 = a1

1 − del1 1

3 +

del1

1 + 134

81

del1 (A10)

del0 = temp

a02 (A11)

a0dp = a0 Re

(1 − del0) (A12)

where a0dp is the mean semi-major axis in kilometers. This

value is entered into the first location of the Brouwer element

array.

For the sample NOAA-12 element set listed in Table A1,

n = 14.22063660 revolutions per day, i = 98.6941◦ and e =

0.0013736. In this example, the equations above give a mean

semi-major axis of 7,192.074 km.

Glossary

AVHRR A dvanced Very High Resolution Radiometer

BER Bit Error Rate

CZCS Coastal Zone Color Scanner

DCF Data Capture Facility

EUVE Extreme Ultraviolet Explorer

GPM General Perturbations Model

GPS Global Positioning System

GSFC Goddard Space Flight Center

G/T System Gain/Total System Noise Temperature

GTDS Goddard Trajectory Determination System

HRPT High Resolution Picture Transmission

IAU International Astrophysical Union

LANDSAT Land Resources Satellite

NASA National Aeronautics and Space Administration

NAVSPA SUR Naval Space Surface Surveillance

NIMBUS Not an acronym, but a series of NASA experimental weather satellites containing a wide variety of atmosphere, ice, and ocean sensors.

NOAA National Oceanic and Atmospheric Administration

NORAD North American Air Defense (Command)

OSC Orbital Sciences Corporation

SeaWiFS Sea-viewing Wide Field-of-view Sensor

SPM Special Perturbations Model

TBUS Not an acronym, but a NOAA orbit prediction

message.

TIROS Television Infrared Observation Satellite

WFF Wallops Flight Facility

Symbols

a0 Intermediate perturbation correction variable.

a1 Orbital semi-major axis in units of Earth radii.

a0dp The mean orbital semi-major axis in kilometers.

D Orbit position difference vector.

Dat Along-track position difference.

Dct Cross-track position difference.

Drad Radial position difference.

del0, del1 Intermediate perturbation correction variables.

e Orbital eccentricity.

Ge Gravitational constant of the Earth (398,600.5

km3 s−2).

i Orbital inclination.

J2 The J2 gravity field term (0.0010863).

J3 The J3 gravity field term (−0.0000254).

J4 The J4 gravity field term (−0.0000161).

J5 The J5 gravity field term.

n Mean orbital motion in revolutions per day.

O Orbit normal vector (P × V ).

P Orbit position vector.

Re Mean radius of the Earth (6,378.137 km).

temp Temporary perturbation correction variables.

V Orbit velocity vector.

xke Revised gravitational constant in units of Earth

radii.

xno Mean orbital motion in radians per minute.

References

Brouwer, D., 1959: Solution of the problem of artificial satellite

theory without drag. Astron. J., 64, 378–397.

Cappellari, J.O., C.E. Velez, and A.J. Fuchs, 1976: Mathematical Theory of the Goddard Trajectory Determination

System. GSFC Report X-582-76-77, 596 pp.

Hoots, F.R., and R.L. Roehrich, 1980: Models for Propagation

of NORAD Element Sets, Project Spacetrack Report No.

3,, Aerospace Defense Command (USAF), 100 pp.

Kidwell, K.B., 1991: NOAA Polar Orbiter User’s Guide

(TIROS-N, NOAA-6, NOAA-7, NOAA-8, NOAA-9,

NOAA-10, NOAA-11, NOAA-12), NOAA/NESDIS

(NCDC/SDSD), Washington, D.C., 279 pp.

Lyddane, R.H., 1963: Small eccentricities or inclinations in the

Brouwer theory of the artificial satellite. Astro. J., 68,

555–558.

16
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 SeaWiFS Technical Report Series

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 Unclassified Unclassified Unclassified Unlimited

Orbit Propagation Models, General Perturbations Model, Special Perturbations Model, 16

Brouwer-Lyddane Model, Along-Track Propagation Model, Cross-Track Propagation

Model

Frederick S. Patt, Elaine R. Firestone, and A.W. Indest:: General Sciences Corporation, Laurel, Maryland;

Charles M. Hoisington: Science Systems Applications, Inc., Lanham, Maryland.

Report is available from the Center for AeroSpace Information (CASI),

7121 Standard Drive, Hanover, MD 21076–1320; (301)621-0390

Unclassified–Unlimited

Subject Category 48

13. ABSTRACT (Maximum 200 words)

National Aeronautics and Space Administration

Washington, D.C. 20546–0001 TM–104566, Vol. 11

Laboratory for Hydrospheric Processes

Goddard Space Flight Center 93B00097

Greenbelt, Maryland 20771

6. AUTHOR(S)

 Frederick S. Patt, Charles M. Hoisington, Watson W. Gregg, and Patrick L Coronado

 Series Editors: Stanford B. Hooker and Elaine R. Firestone

 Technical Editor: A.W. Indest

An analysis of orbit propagation models was performed by the Mission Operations element of the Sea-viewing Wide

Field-of-view Sensor (SeaWiFS) Project, which has overall responsibility for the instrument scheduling. The orbit

propagators selected for this analysis are widely available general perturbations models. The analysis includes both

absolute accuracy determination and comparisons of different versions of the models. The results show that all of the

models tested meet accuracy requirements for scheduling and data acquisition purposes. For internal Project use the SGP4

propagator, developed by the North American Air Defense (NORAD) Command, has been selected. This model includes

atmospheric drag effects and, therefore, provides better accuracy. For High Resolution Picture Transmission (HRPT)

ground stations, which have less stringent accuracy requirements, the publicly available Brouwer-Lyddane models are

recommended. The SeaWiFS Project will make available portable source code for a version of this model developed by

the Data Capture Facility (DCF).