Astrodynamics 101 - DerAstrodynamics.com

Astrodynamics 101 

Part 1: Kepler+ Algorithm 

J. P. Vinti, “Orbital and Celestial Mechanics” 

AIAA Book, V177, 1998 

object at t 2 

v (t ) 

2 

m i 

m 

k 

Kepler Problem 

Given: r ( t ) , v ( t ) , t , t 

Find: r ( t ) , v ( t ) 

r (t ) 

2 

m 

j 

Sphere / 

Spheroid 

2 

1 

+ 

2 

Kepler 

Predicted 

Orbit 

2-Body 

Predicted 

Orbit 

99+% 

1 

v (t ) 

1 

Gim Der 

DerAstrodynamics 

July 5, 2013 

1 

mN 

object 

at t 

1 

r (t ) 

1 

2

Analytic Astrodynamics Overview 

Astrodynamics 101: Kepler+ Algorithm 

Part1: Analytic Prediction Algorithms 

Part2: Verifications 

Astrodynamics 102: Lambert+ Algorithm 

Part1: Analytic Multi-revolution Targeting Algorithms 

(Orbit Determination for Radar Data) 

Part2: Verifications 

Astrodynamics 103: Gauss/Laplace+ Algorithm 

Part1: Analytic Angles-only Algorithms 

(Orbit Determination for Optical Sensor Data) 

Part2: Verifications

Astrodynamics 101: Kepler+ Algorithm 

Part 1. Analytic Prediction Algorithms 

1. What, Why, How 

2. Physics 

3. Equations of motion 

4. Analytic Algorithms for Prediction 

5. Applications for SSA 

Part 2. Verifications 

6. Matter of Reference 

7. Numerical Examples

Satellites, Missiles and Debris 

Intentional Destruction and Unintentional Space Collisions 

2007 2009, . . , 2013 

ASAT Test and Space Accidents Happened

Destruction Power of Space Debris 

Properties of Space Debris (2010): 

Size: Trackable ~ 10 cm (4 in) 

speed: > = 10 km/s (+22,000 mph) 

Range: 600 to 1600 km (400 to 1000 mi) 

Distribution: 7% useful satellites, others junk 

The image above shows, the risk of damage is real. This hole over 1 cm (3/8 in) 

in diameter penetrates the Hubble high gain antenna dish (the unit continued 

working in spite of the damage). 

The windows on the Space Shuttles have been replaced 80 times due to 

impacts with objects of less than 1 mm (0.04 in). And costly systems to track 

and issue daily emails warning of potential impacts must be maintained.

Destruction Power of Space Debris 

All it took to punch this 0.025-cm hole in a U.S. satellite was a paint chip moving at hypervelocity 

of 9 km/s. When the shuttle brought back the satellite, scientists found six holes per square foot. 

(Hole in satellite photograph by NASA.)

How Many? Then (2009) 

http://www.ucsusa.org/assets/documents/nwgs/SatelliteCollision-2-12-09.pdf 

Size of debris particles 10 cm (4 inches) 

(Can be Tracked) 

Debris in LEO and HEO 

Mostly 600 to 1500 km 

14,000 

(Space Fence 

Targets) 

1 cm (0.4 inch) 

(Cannot be 

Tracked) 

370,000 

Debris at all altitudes 22,000 750,000 

Total estimated debris, by size, in orbit around the earth (26 Feb 2009)

How Many? Now (2013) 

http://www.space.com/12602-space-junk-cleanup-grand-challenge-21st-century.html 

Total estimated debris, by size, in orbit around the earth (8 Mar 2013)

How Debris Clouds Look Like? 

Debris clouds after 9 minutes Debris clouds after 10 days 

Debris Clouds after 6 months 

Debris clouds after 3 years 

Debris Generation After a Collision

GS satellite ring 

alt. = ~1400 km 

rev/day = ~ 12 

Conjunction Assessment with Less False Alarms 

Real Time and Accurate 

Conjunction Assessment 

Iridium satellite ring 

alt. = ~800 km 

rev/day = ~ 14 

Less False Alarms

Orbit Determination of All Space Objects for SSA 

90% of the Space Objects in Near-Earth 

from 22,000 to 100,000 soon 

Kepler +, Lambert +, Gauss/Laplace+ Algorithms 

Fast and Accurate Analytic Astrodynamics Algorithms 

for Precise Orbit Determination

1. What, Why, How 

What? Cataloging of 100,000+ satellites and 

space debris is a challenging Space 

Situation Awareness (SSA) problem 

Why? Need near real time positions and 

characterization of all objects to insure 

operation and safety of space assets 

How? Understand the Physics and develop 

new analytic Astrodynamics algorithms 

for accurate and fast trajectory prediction 

and SSA applications 

Building A New Space Catalog of 100,000+ Objects 

Requires Analytic Kepler+ Algorithm

object 

v (t 

2 

) at t 

2 

r (t ) 

2 

Sphere 

2-Body 

Predicted 

Trajectory 

r (t ) 

1 

v (t ) 

1 

object 

at t 1 

Kepler and Kepler+ Algorithms 


Given: t , t , r ( t ) , v ( t ) 

1 

Find: r ( t ) , v ( t ) 

2 

2 

add perturbations analytically 

( J 2 , J 3 , J 4 , J 22 , J 31 , . . 

Sun , Moon , Drag , . . ) 

Kepler (2-Body) solution Kepler+ solution for SSA 

(accurate and fast ) 

2 

1 

1 

v (t ) 

2 

Spheroid 

object 

at t2 

r (t ) 

2 

Kepler+ 

Predicted 

Trajectory 

r (t ) 

1 

v (t ) 

1 

object 

at t 1

Forces and Accelerations 

2. Physics

v (t ) 

object at t 

r (t) 

Accelerations on Satellite and Rocket (1) 

1. Earth spherical gravity (low_g) 

2. Earth spheroidal gravity (high_g) 

3. Sun gravity 

4. Moon gravity 

5. Solar radiation pressure 

6. Atmospheric (Air) drag 

7. Thrust acceleration 

Earth 

v ( t ) 

1 

object at t 

1 

r ( t 

1 

) 


Given: r (t ) , v (t ) , t , t 

1 

Find: r (t) , v (t) 

1 1

1. Spherical gravity (low_g) 

v (t ) 

object at t 

r (t) 

 

a = r 

r 3 

low 

Example: 

MEO / GEO object at t 

r = 7,000 / 40,000 km 


Spherical 

Earth 

a = 8.0 0.2 

low 

v (t ) 

m/s 2 

2. Spheroidal gravity (high_g) 

object at t 

r (t) 

a = a + a + a + . . + 

high 

J 2 

J 3 

J 4 

+ a + . . . + 

J 

22 

Oblate Spheroidal 

'Flattened' Earth 

3 

a = 20 x 10 1.0 x 10 

high 

6 

m/s 2


a = r 

sun 

v (t ) 

object at t 

r (t) 

Earth 


Sun 

 

r (t) 

sun 

sun 

r 3 

 

a = r 

moon 

r moon 

3 

sun moon 

v (t ) 

object at t 

r (t) 

Earth 

Example: 

MEO / GEO object at t sun moon 

r = 7,000 / 40,000 km m/s 2 

6 

a = 0.3 x 10 

6 

2. x 10 


r (t) 

moon 

6 

a = 0.6 x 10 

5. x 10 

Moon 

m/s 2 6

5. Solar Radiation Pressure 

a (t) 

solar 

radiation 

pressure 

v (t ) 

object at t 

r (t) 

Earth 

Example: 

MEO / GEO object at t 

r = 7,000 / 40,000 km 


Sun 

Photons 

r (t) 

sun 

solar 

radiation 

pressure 

Earth 

6 

a = 0.1 x 10 m/s 2 

Atmo 

6. Atmospheric (Air) Drag 

sphere 

Earth 

a = 0 

drag 

v (t ) 

r (t) 

m/s 2 

Earth 

Atmosphere 

Air 

a (t) 

drag 

effective height 

for drag: 

Sat. = ~ 200 km 

Missile 

~ 

= 60 km 

(for short time 

prediction)

V 

Transfer 

Orbit 

v 

t1 

1 

v 

1 

t 1 

t 

2 

r 

1 

r 

2 

Earth 

v 

t2 


Orbit 1 

Orbit 2 

V 

v 

2 

Satellite Thrusting 

(Orbit Change or Maneuver) 

2 

7. Thrust Accelerations 

Lambert Problem 

Given: r , r , t , t 

1 2 1 2 

Find: v (t ) = v , v (t ) = 

1 

Then compute: 

V = v 

1 

t1 

t1 

v 

1 

V = v v 

2 2 t2 

2 

v 

t2 

Rocket Thrusting 

(Ascending from Earth)

7 

V 

v (t ) 

object at t 

2 

4 

Protons 

r (t) 

1 

Air 

6 

Moon 

5 

Earth 

r (t ) 

1 

Accelerations Summary (6) 

v (t ) 

1 

object 

at t 1 

3 

Sun 

1. Earth low gravity 

2. Earth high gravity 



5. Proton Pressure 

6. Air Drag 

7. Thrust Acceleration 


Given: r (t 

1 

) , v (t ) , t , t 

1 1 

Find: r (t) , v (t) 

(Pull) 

(Pull) 

(Pull) 

(Pull) 

(Push) 

(Pull) 

(Push)

3. Equations of Motion 

Equations 

for 

Prediction and Propagation

Mathematical Interpretation 

Astrodynamics: Understanding Forces 

Newton’s Formula 

General Formula: 

Orbiting Satellites: 

Missiles / Aircraft: 

m 

m 

m 

2 

d r d r 

= f ( t , r , ) = F 

2 

d t 

d t 

2 

d r 

d t 

d t 

2 

2 

d r 

r 

= f ( ) = F + F 

3 

gravity 

r 

others 

= F + F + F + F 

2 gravity thrust aero wind

Equations 

of Motion 

Formulation 

Who 

Analytic 

Solution 

Force Method 

2 

d r 

dt 

2 

= 

Classical 

(Newton) 

 

Kepler, Newton, 

(almost everyone) 

Keplerian 

(2-body: a d = 0 ) 

r 

3 

Comparison of Analytic Methods 

r a 

d 

p 

k 

H( 

q, 

p, 

t) 

, 

q 

Energy Method 

k 

Von-Zeipel, Laplace, (Vinti, 

(Brouwer, SGP4) Kepler+) 

non-Keplerian non-Keplerian 

(averaging (general 

solution: a 

d 

0 ) solution: a 

d 

0 ) 

q 

Hamilton - Jacobi 

AIAA book V177, “Orbital and Celestial Mechanics” 

k 

H( 

q, 

p, 

t) 

 

p 

k

Newton / Classical Formulation 

Equations of motion 

2 

d r 

= r a 

2 3 d t r 

Given : r(t 0), v(t 

0), t 0 , t 

Find : r(t ) , v(t 

) 

Kepler method: 

Assume zero perturbation: 

, 

a 0 

d = 

resulting in spherical 

gravity effect. 

Kepler analytic solution: 

r fI = 

v 

fI gI 

gI 

 

 

r0 

v 

where f , g, f , and g are 

analytic and functions of 

r r(t ) and v v(t 

) 

 

 

 

d 

0 

0 0 0 0 

UNCLASSIFIED 

Comparison of Analytic Solutions 

Vinti / Hamilton-Jacobi Formulation 

Equations of motion 

dpk H(q, p, t ) dqk 

H(q, 

p, t ) 

, 

d t qk d t pk 

where q's and p's are respectively 

coordinates and momenta, and k 1, 

2, 3 

Given : r(t 0), v(t 

0), t 0 , t 

Find : r(t ) , v(t 

) 

Vinti method: 

Assume non-zero perturbation: 

, 

a 0 

d 

resulting in spheroidal gravity effect so that 

zonal geopotentials J , J , J are included. 

2 3 4 

Vinti analytic solution: 

See Vinti AIAA book, V177, 1998, 

"Orbital and Celestial Mechanics", 

Chapter 8. 

Also see GTDS, Chapter 5.12 on 

attributes of "Vinti Theory", singularity free 

and applicability to satellites and missiles.

4. Analytic Algorithms for Prediction 

Analytic Algorithms 

and 

Developement

Analytic Algorithms and Solutions 

Important Questions: 

1. How accurate? 2. How fast? 3. How robust? 

Answers: 

1. Accuracy of 10 km to a few centimeters depends 

on SSA applications of prediction / tracking / 

correlation / conjunction / . . . . 

2. Need to process all objects in the current or future 

Space Catalog under 12 hours or much quicker 

3. Robustness requires algorithms to update everyday 

accurately all objects in any Space Catalog of tens 

or hundreds of thousands objects

Kepler Equations of Motion (2-Body) 

2 

d r 

= r 

2 3 d t r 

Kepler Algorithm (2 Steps for solution) 

1. Solve Classical or Universal Kepler Eqn 

for E or x by an iterative method such as 

Newton, Halley or Laguerre 

F(E) = E e sinE 0 

F (x) = (x) (t 2t 1) 

0 

2. Compute (t ) and 

r(t 2) fI gI 

r(t 

1) 

= 

v(t ) fI gI 

v(t 

) 

 

 

 

 

 

2 1 

where f g 

r 2 v 2 

(t ) 

, f g are functions of E or x 

, , 

Analytic Kepler Algorithm 


Given: t , t , r ( t ) , v ( t ) 

1 

Find: r ( t ) , v ( t ) 

v (t 

2 

) 

r (t ) 

2 

2 

object 

at t 2 

2 

2 

1 

Sphere 

Predicted 

Trajectory 

1 

r (t ) 

1 

v (t ) 

1 

object 

at t 

1


Given: r (t 1) , v (t ) , t , t 

1 1 2 

Find: r (t 

2 

) , v (t 

2 

) 

v (t 

2 

) object at t 

2 

r (t ) 

2 

Exaggerated 

Spheroid 

>> 1% mass 

~ 0.001 m/s**2 

acceleration 

Vinti 

Predicted 

Trajectory 

v (t ) 

1 

r (t ) 

1 

object 

at t 1 

Analytic Vinti Algorithm 

Vinti Implementation (AIAA Book) 

1. Hamilton-Jacobi Equations of motion 

2. Transform given ECI state 

to Spheroidal coordinates 

3. Include J 2 , J 3 , J 4 and solve for 

final state in Spheroidal coordinates 

4. Transform back to final ECI state 

Vinti Theory 

(conceptual Equations of motion) 

2 

d r 

= r a a a a 

2 3 

dt R 

Spheroidal 

Body 

J2 J3 J4 Vinti 

Non-Keplerian 

(more accurate) 

Trajectory

+ 

Kepler solution 

(Vinti and other perturbations) 

Numerical accurate 

solution (desired) 

V (t ) 

General Equations of motion: 

d2 

R 

= 

 

R a 

2 3 d t R 

d 

Analytic Kepler+ Algorithm 

(3) 

Vinti spheroidal 

solution only 

R (t) 

General method of solution 

=> Kepler+ solution 

(1) Kepler solution only 

(2) Kepler + a d perturbed solution 

(3) Vinti spheroidal solution only 

, 

a d = 

0 

(2) Kepler solution with a 

(other perturbations) 

Central 

Body 

(1) 

R (t ) 

1 

Kepler 

solution only 

V (t ) 

1 

object at t 1 

d

Vinti and Kepler+ History 

Vinti developed theory and algorithm in 1959~1970s 

vinti1 – Wadsworth, rocket free-flight, 1963 

vinti2 – Izsak-Borchers, ICBM onboard targeting, 1965 

vinti3 – Bonavito, Goddard trajectory s/w, 1966 

vinti4 – Lang, MIT MS thesis under Vinti, 1968 

vinti5 – Getchell, military satellite analysis at NSA, 1970 

vinti6 – Der-Monuki, satellite and missile trajectories, 1998 

vinti7 – Der, Satellite and Missile trajectories, no singularities, 2009 

Kepler+ = Vinti7 + additional perturbations, 2012 

Includes analytically additional perturbations of Higher-order 

Geopotentials, Sun, Moon, Drag, . . , for Satellites and Missiles 

Provides Speed, Robustness and “Near SP” Accuracy needed 

for SSA 

The most complete General Perturbation algorithms

APL Technical 

Digest Vol 27, #3 

(2007) 

Revisit Key Astrodynamics Algorithms 

Vinti 

Theory (1959) 

+ 3 22 

rd Kepler+ = 

+ J 2 , + J 3 , + J 4 , 

body, J , J 31 , 

drag, . . (2012) 

No funding for accurate and 

fast analytic Astrodynamics 

algorithms since 1960s! 

Spent 50 years funding accurate 

but slow Astrodynamics algorithms 

that are not efficient nor suitable for 

SSA 

References: SSA papers 

• Desrocher,D., “Transforming Space Surveil..”, AAS 05-201 

• Morring, F. Jr., “Collision Course”, Aviation Week, March 19, 2012 

• Vetter, J.R., “Fifty Years of Orbit Determ . . , APL TD, V27, #3 

• . . . . .

Osculating Orbital 

Elements at t o 

[ r (t ), v (t )] 

o o 

Radars / 

Optical 

sensors 

Future 

look angles 

( pointing 

prediction ) 

Others . . . . . . 

Orbit Determination and Prediction/Propagation 

Object at t 

r 

Unsuitable for real-time 

automatic processing 

Ephemerides 

(Catalogs) 

Raw 

Observation 

data 

v 

Orbit 

Determination 

(Estimated future/past) 

position and velocity vectors 

Rise/Set 

Site visibility 

Astrodynamics 

102: Lambert / 

103: Gauss/Laplace 

Initial Orbit 

Determination 

Processed 

Observation 

data 

Differential 

Correction 

Batch / KF 

TLE 

conversion 

difficulties 

Singularity 

difficulties 

Osc2Mean 

Analytic algorithms in OD and P/P 

need to process 100,000+ Objects 

in less than 12 hours with accuracy 

of 10 km to centimeters for SSA 

(Estimated initial) 

positin and 

velocity vectors 

Orbital 

element set 

r and v 

Prediction / 

Propagation 

1 2 

SP 

3 

Numerical 

SGP4 

Integration Kepler+ 

Close approach 

(miss distance) 

Osculating Orbital 

Elements at t 

[ r , v ] 

Astro 101: 

Kepler 

1 

2 

3 

SGP4 needs TLE 

conversion 

(not efficient for SSA) 

SP is accurate, but 

slow for SSA 

Kepler+ is accurate 

and fast for SSA

Key SSA Features of Kepler+ 

SGP4 Kepler+ 

Prediction Speed (~17 / 14+ micro-sec per traj) YES YES 

Prediction error growth ( ~5 / 1 km per day) YES YES 

Perturbations compliant and analytic YES YES 

Same input/output osculating vectors as SP NO YES 

Use any initial state vector as SP NO YES 

Use new updated state vector and time as SP NO YES 

Singularity free for any object or orbit as SP NO YES 

Flexibilities for satellite, missile, Earth/Sun, . . NO YES 

SP = numerical integration

5. Applications of Analytic Algorithms 

SSA Applications

Atm. Drag effects 

SS, 

ISS 

Applications of Analytic Algorithms 

Iridium 

LEO + MEO 

789 – 865 km 

Kepler+ 

to GEO 

and 

beyond 

Cataloged object distribution vs. altitude from LEO to HEO 

after the Accidental Collision on February 10, 2009. 

Globalstar 

Sun & Moon effects

SF 

Example: I = J = K = 100 

Correlation combination 

= I J K = 1,000,000 

Takes a few seconds for 

a million combinations 

UNCLASSIFIED 

Applications of Analytic Algorithms(2) 

Multi-sensor Multi-object UCT Cataloging using Radar Data 

Fence or Radar 

Correlating 

90+ % of 

objects to 

catalog 

SF 

J 

UCT processing 

Solve by New 

angles-only algorithm: 

3 computed ? 

ranges = 

I 

K 

3 detected 

ranges

UNCLASSIFIED 


Multi-sensor Multi-object UCT Cataloging using Optical Sensor Data

Problem: SSA 


Solution: Analytic Astrodynamics

Analytic Algorithm Pioneer -- Prof. Brouwer 

Professor Dirk Brouwer lecturing on the motion of the Moon

Analytic Algorithm Pioneer -- Prof. Vinti 

Professor John Vinti lecturing on Potential Theory


Part 2: Verifications of Prediction Algorithms 

Space Situation Awareness (SSA) 

Uncorrelated Target (UCT) Cataloging 

Satellite and Debris Conjunction 

and Collision 

Satellite and Missile Prediction 

and Propagation 

Orbit Determination 

. . . 

J. P. Vinti, “Orbital and Celestial Mechanics” 

AIAA Book, V177, 1998 

Gim Der 

DerAstrodynamics


Part 2: Verifications for SSA 

(Also please download iOrbit: 

http://derastrodynamics.com/index.php?main_page=index&cPath=1_7 

and run kep for Astrodynamics 101 Verifications) 

Next

Reference Trajectory (General) 

6. Matter of Reference 

• Numerical integration with perturbations (Slide 15 to 21) 

gives analytic solutions with kilometer accuracy 

• Numerical integration with additional perturbations 

(time and coordinate, tides, J [C and S ], planets, . . . , ) 

gives numerical solutions with centimeter (or less) accuracy 

Reference Trajectory (Section 7. Numerical Examples) 

• Numerical integration with perturbations 

(WGS84 12x12, Sun, Moon, Drag) 

21 21 21 

Comparison (Algorithms and Criteria) 

• Kepler, SGP4, Vinti, Kepler+, numerical integration with 

perturbations (WGS84 4x4, Sun, Moon, Drag) 

• CPU timing, accuracy, robustness, mean, standard deviation, max

UNCLASSIFIED 

7. Numerical Examples 

Numerical Examples 

of 

Geocentric Objects

Numerical Example 1 

Kepler_Test1 * (Analytic Kepler algorithms compared with Numerical Integration) 

High Earth Orbit (HEO) , typical 2-Body solutions using the 2007 TLE file 

Input: (2007_249, isat = 28836) 

t1 = 0., t2 = 86400. (seconds) 

Output: kepler1a failed to converge, kepler1b and kepler2 converged to correct answer 

algorithms r_eci (t2) (km) v_eci (t2) (km/s) 

kepler1a 53052.0539 54034.8022 30994.6668 

kepler1b 27456.2663 16961.1113 15098.0763 

kepler2 27456.2663 16961.1113 15098.0763 

Num. Int. 

(2-Body) 

r_eci (t1) (km) v_eci (t1) (km/s) 

2218.922362 13049.809380 15.285102 3.894738598 4.630875574 2.332314347 

* 

27456.2662 16961.1114 15098.0764 

6.451059 4.143735 2.850977 

1.431551 1.341058 0.596545 

1.431551 1.341058 0.596545 

1.431551 1.341058 0.596545 

Free download: Kepler_Test computer programs with source code


Kepler_Test1 * (Analytic Kepler algorithms compared with Numerical Integration) 

High Earth Orbit (HEO) , typical 2-Body solutions using the 2009 TLE file 

Input: (2009_298, isat = 19773) 

t1 = 0., t2 = 86400. (seconds) 

Output: kepler1b failed to converge, kepler1a and kepler2 converged to correct answer 

DerAstro r_eci (t2) (km) v_eci (t2) (km/s) 

kepler1a 16906.0049 1871.5115 2495.6926 4.125426 3.558097 0.604363 

kepler1b 99927.4767 92921.1426 14631.3162 5.642214 7.607469 0.840981 

kepler2 16906.0049 1871.5115 2495.6926 4.125426 3.558097 0.604363 

Num. Int. 

(2-Body) 


209.011468 30385.859968 4.280644 2.248716032 2.177749649 0.329155365 

* 

16906.0051 1871.5115 2495.6927 4.125426 3.558097 0.604363 

Free download: Kepler_Test computer programs with source code


Analytic Kepler, SGP4, Vinti, Kepler+ algorithms and num. int. (wgs 4x4, . . .) 

compared with num. int. (wgs 12x12, . . .) 

Low Earth Orbit (LEO) , typical 2-Body and perturbed solutions using the 2009 TLE file 

Input: (2009_249, isat = 4382) 

t1 = 0., t2 = 86400. (seconds) 

Algorithm r_eci (t2) (km) v_eci (t2) (km/s) 

kepler2 346.687092 6795.012489 412.914501 

sgp4 257.913989 6719.684754 1214.516634 

vinti 207.380964 6701.510662 1337.255798 

kepler+ 213.326456 6703.763322 1322.957867 

num. int. 

(4x4, . .) 

num. int. 

(12x12, . .) 


510.306735 6793.878710 0.007287 2.949065600 0.256823630 7.486987267 

Output: kepler2 is poor. sgp4 is fair. vinti is reasonable. 

kepler+ and num. int. (4x4, . .) produced good solutions for SSA purposes. 

213.622528 6703.908820 1322.028866 

216.109788 6704.762171 1316.193868 

2.978862 0.215630 7.472634 

3.040136 1.016065 7.361291 

3.045054 1.156390 7.334489 

3.044552 1.140038 7.337720 

3.044515 1.139009 7.337933 

3.044353 1.132308 7.339295


Analytic Kepler, SGP4, Vinti, Kepler+ algorithms and num. int. (4x4, . . .) 

compared with num. int. (12x12, . . .) 

Low Earth Orbit (LEO) , typical 2-Body and perturbed solutions using the 2009 TLE file 

Input: (2009_249, isat = 23282) 

t1 = 0., t2 = 86400. (seconds) 

Algorithm r_eci (t2) (km) v_eci (t2) (km/s) 

kepler2 5496.729940 3059.667485 3715.519008 

sgp4 5233.338371 2920.278629 4171.411336 

vinti 5240.654526 2919.268575 4163.061193 

kepler+ 5248.036696 2918.287891 4154.564282 

num. int. 

(4x4, . .) 

num. int. 

(12x12, . .) 


6969.687412 2195.624199 0.030412 0.695626778 2.267918155 6.958092777 

Output: kepler2 is poor. sgp4 is fair. vinti is reasonable. 

kepler+ and num. int. (4x4, . .) produced good solutions for SSA purposes. 

5248.183301 2918.362041 4154.359585 

5250.523888 2917.968463 4151.680633 

4.385493 0.712399 5.858270 

4.806832 0.629942 5.529725 

4.798649 0.634552 5.536259 

4.790380 0.639186 5.542832 

4.790136 0.639259 5.543012 

4.787513 0.640780 5.545121

Algorithm 

SGP4 

(analytic) 

Vinti 

(analytic) 

Kepler+ 

(analytic) 

Numerical 

Integration 

(4x4, 200s) 

Comparison of CPU timing, Perturbations, Singularity 

2007 (Sep 6) 

11140 objects 

CPU timing (micro-seconds) 

per trajectory 

18.2 

14.0 

8,400. 

127,000. 

2009 (Oct25) 

14327 objects 

17.4 

14.2 

8,000. 

127,000. 

2011 (Jan 5) 

14638 objects 

17.4 

13.9 

7,800. 

127,000. 

Perturbations 

J 2 , J 3 , J 4 

Sun, Moon, Drag 

J 2 , J 3 , J 4 

J 2 , J 3 , J 4 , 

J 22 , J 31, 

J 32 , 33 

J , 


WGS84 4x4 


Singularity 

Yes 

No 

No 

No

Algorithm 

(Reference = 

WGS84 12x12, 

Sun, Moon, 

Drag) 

SGP4 

(analytic) 

Vinti 

(analytic) 

Kepler+ 

(analytic) 

Numerical 

Integration 

(4x4, 200s) 

Comparison of Robustness, Mean and Standard Deviation 

2007 

total # of objs = 11140 

# of objects 

with position mean 

difference / std (km) 

> 5 km 

7043 

4129 

1117 

131 

Robustness, mean and standard deviation (std) 

for one day prediction 

53. / 421. 

5.3 / 5.0 

2.3 / 1.9 

1.5 / 1.4 

2009 


# of objects 



> 5 km 

9028 

5399 

1964 

216 

52. / 257. 

5.4 / 5.1 

2.6 / 2.1 

1.7 / 1.4 

2011 


# of objects 



> 5 km 

9289 

5453 

1896 

177 

46. / 195. 

5.5 / 5.3 

2.6 / 2.0 

1.6 / 1.4

Algorithm 

(Reference = 

WGS84 12x12, 

Sun, Moon, 

Drag) 

SGP4 

(analytic) 

Vinti 

(analytic) 

Kepler+ 

(analytic) 

Numerical 

Integration 

(4x4, 200s) 

Comparison of Robustness and Max Error wrt Reference 

Robustness and max error with respect to Reference 

for one day prediction 

2007 


# of objects 

with position max error 

difference (km) 

> 10 km 

4767 

2093 

25 

7 

37828. 

84. 

32. 

20. 

2009 


# of objects 



> 10 km 

6243 

2727 

39 

11 

13138. 

128. 

38. 

25. 

2011 


# of objects 



> 10 km 

6161 

2912 

32 

6 

6025. 

73. 

31. 

15.

UNCLASSIFIED 



Of 

Geocentric Objects 

(More examples available upon request)

UNCLASSIFIED 



Of 

Heliocentric Objects 

(Available upon request, but Not for SSA)

Astrodynamics 101 - DerAstrodynamics.com

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?