A Brief Introduction to Astrodynamics

Shaun Gorman

Iowa State University

Ames, Iowa

Topics Discussed

• Coordinate Systems

• Orbital Geometry

• Classical Orbital Elements

• Classes of Orbits

• Two-line Element Sets

Coordinate Systems

• Heliocentric-Ecliptic Coordinate System

• Geocentric-Equatorial Coordinate System

• Right Ascension-Declination System

• Perifocal Coordinate System

Heliocentric-Ecliptic Coordinate System

• Origin at the center of the sun.

• X-Y plane coincides with the earth’s plane of revolution

• X axis points in the direction of the vernal equinox

• Z axis points in the direction of the suns north pole

Geocentric-Equatorial Coordinate System

• Also called Earth Centered Inertial or ECI• Origin at the center of the earth• X-Y plane coincides with the earth’s equator• X axis points in the direction of the vernal equinox• Z axis points in the direction of the north pole• I, J and K unit vectors lie along the X, Y and Z

axes

Right Ascension-Declination System

Perifocal Coordinate System

• Origin at the center of the earth• P-Q plane coincides with the satellites orbit

plane• P axis points in the direction of the vernal

equinox• Q axis is 90o from the P axis in the direction

of satellite motion• W axis is normal to the satellite orbit

ECI Coordinate Systems

• Several different types of ECI coordinate systems.– Fixed– J2000– B1950– TEME of Epoch– TEME of Date

ECI Coordinate Types

Classical Orbital Elements (COE)

Uses the traditional osculating Keplerian orbital elements to specify the shape and size of an orbit.

Cartesian Uses the initial X, Y and Z position and velocity components of the satellite.

Fixed

• X is fixed at 0 deg longitude, Y is fixed at 90 deg longitude, and Z is directed toward the north pole.

• Only Cartesian type of coordinates can be used.

J2000

• X points toward the mean vernal equinox and Z points along the mean rotation axis of the Earth on 1 Jan 2000 at 12:00:00.00 TDB, which corresponds to JD 2451545.0 TDB.

• Can use either Cartesian or COE.

B1950

• X points toward the mean vernal equinox and Z points along the mean rotation axis of the Earth at the beginning of the Besselian year 1950 (when the longitude of the mean Sun is 280.0 deg measured from the mean equinox) and corresponds to 31 December 1949 22:09:07.2 or JD 2433282.423.

• Can use either Cartesian or COE.

TEME of Epoch

• X points toward the mean vernal equinox and Z points along the true rotation axis of the Coordinate Epoch.

• Can use either Cartesian or COE.

TEME of Date

• X points toward the mean vernal equinox and Z points along the true rotation axis of the Orbit Epoch.

• Can use either Cartesian or COE.

Orbital Geometry

• Apoapsis- farthest point in an orbit

• Periapsis- nearest point in an orbit

• Line of Nodes - The point where the vehicle crosses the equator

• Radius - distance from the center of the Earth to the orbit

Orbital Geometry

Classical Orbital Elements

a - Semi-major Axis-a constant defining the size of the orbit

e – Eccentricity-a constant defining the shape of the orbit (0=circular, Less than 1=elliptical)

– Inclination-the angle between the equator and the orbit plane

- Right Ascension of the Ascending Node-the angle between vernal equinox and the point where the orbit crosses the equatorial plane

- Argument of Perigee-the angle between the ascending node and the orbit's point of closest approach to the earth (perigee)

v - True Anomaly-the angle between perigee and the vehicle (in the orbit plane)

C.O.E. (continued)

Vector Re-fresher

• Before we start lets go over some basic vector math

KJI

KJI

ba

a

KJII

KJIa

JIJIKIKIKJKJ

KJI

KJI

2K

2J

2I

KJI

abbaabbaabba

bbb

aaa

aaaa

001

aaa

Determining Orbital Elements

• Let’s say that a ground station on the earth is able to provide the position and velocity of a satellite by providing us with vectors r and v.

Conversion from Cartesian to COE

• Given the position and velocity vectors: r and v

• Determine the six classical orbital elements: e, a, i, , and v

Setting up a coordinate system

• We will use the geocentric equatorial coordinate system.

• The I axis points towards the vernal equinox.

• The J axis is 90o to the east in the equatorial plane.

• The K axis points directly through the north pole.

Determining Orbital Elements

• The expression, which is called specific angular momentum, must be held constant due the law of conservation of angular momentum.

• Thus:

KJI

KJI

h KJI

KJI

KJI hhh

vvv

rrr

vrh

Determining Orbital Elements

• An important thing to remember is that h is a vector perpendicular to the plane of the orbit. The node vector is defined as.

• Thus:

hKn

JhIhKnJnIn

hhh

100

KJI

IJKJI

KJI

n

Determining Eccentricity

• The eccentricity vector is just a function of the gravitational parameter and the r and v vectors

• For the Earth

ee

vr

r

μv

μ

1 2 vre

2

316

sec

ft104076468821 .

Determining Semi-major Axis

• The equation for the semi-major is a function of the velocity and radius vectors along with the gravitational parameter

• If e=1, a=inf.

12

μ

v

r

2a

Determining Inclination

• Since the inclination is the angle between K and h, the inclination can be found using the formula:

• Inclination is always between zero and pi.

h

hcos(i) k

Determining RAAN

• Since the Right Ascension of the Ascending Node is the angle between I and n, the inclination can be found using the formula:

• RAAN is always between pi and two pi.

n

n)cos( I

Determining Argument of Perigee

• Since the Argument of Perigee is the angle between n and e, the inclination can be found using the formula:

• Argument of Perigee is always between zero and pi.

ne)cos(

en

Determining True Anomaly

• Since the True Anomaly is the angle between e and r, the inclination can be found using the formula:

er)cos(

reOv

Classes Of Orbits

• Types of rotation– Prograde

– Retrograde

– Polar

• Types Of Orbital Geometry– Elliptical

– Circular

– Parabolic

– Hyperbolic

Prograde

• The Prograde or direct orbit moves in direction of Earth's rotation

• 0o<i<90o

Retrograde

• The retrograde or indirect moves against the direction of Earth's rotation

• 90o<i<180o

Polar

• Direct orbit over north and south pole

• i=90o

Elliptical

• Eccentricity, 0<e<1

• Semi-major Axis, rp<a<ra

• Semiparameter, rp<p<2rp

Circular

• Eccentricity, e=0• Semi-major Axis, a=r• Semiparameter, p=r

Parabolic

• Eccentricity, e=1• Semi-major Axis, a=inf

• Semiparameter, p=2rp

Hyperbolic

• Eccentricity, e>1• Semi-major Axis, a<0

• Semiparameter, p>2rp

Two-line Element Sets

• One of the most commonly used methods of communicating orbital parameters is the Two-line element sets generated by NORAD. It is important to note that TLEs were developed for use only with the MSGP-4 propagator. Using TLEs with any other propagator may invalidate some of the built-in assumptions.

• These elements contain most of the same elements as the classical orbital elements, along with some additional parameters for identification purposes and for use in modeling perturbations in the MSGP-4 propagator.

TLEs

• TLEs contain 12 different variables– Six for the Classical Orbital Elements

• Four actual C.O.E.s: e, i, and • Two variables that can be used in place of C.O.E.:

, Mean motion and n, mean anomaly

– Three to describe the effects of perturbations on satellite motion: Bstar, and

– Two for identification purposes– One for the time when this data was observed

2

n6

n

TLE format

• The following is an example of a Two-line Element set.

• This Format looks rather intimidating and is read the following way

1 1 6 6 0 9 U 8 6 0 1 7 A 9 3 3 5 2 . 5 3 5 0 2 9 3 4 . 0 0 0 0 7 8 8 9 0 0 0 0 0 0 1 0 5 2 9 - 3 3 4

2 1 6 6 0 9 5 1 . 6 1 9 0 1 3 . 3 3 4 0 0 0 0 5 7 7 0 1 0 2 . 5 6 8 0 2 5 7 . 5 9 5 0 1 5 . 5 9 1 1 4 0 7 0 4 4 7 8 6

Y Y D D D . D D D D D D D D1 1 6 6 0 9 U 8 6 0 1 7 A 9 3 3 5 2 . 5 3 5 0 2 9 3 4 . 0 0 0 0 7 8 8 9 0 0 0 0 0 0 1 0 5 2 9 - 3 3 4

2 1 6 6 0 9 5 1 . 6 1 9 0 1 3 . 3 3 4 0 0 0 0 5 7 7 0 1 0 2 . 5 6 8 0 2 5 7 . 5 9 5 0 1 5 . 5 9 1 1 4 0 7 0 4 4 7 8 6

BstarElement Number

InclinationRight Ascension

of nodeEccentricity

Argument of perigee

Mean Anomaly Mean Motion

Satellite Number

International Designator

Epoch2

n6

n

TLE Classical Orbital Elements

• The two-line element sets provide four of the classical orbital elements : e, i, and

• Instead of true anomaly the TLE gives the mean anomaly because it can be calculated at future time easier.

• This is also true for the substitution of mean motion for semi-major axis which will be explained on the next slide.

Mean Motion to Semi-major Axis

• n=15.5911407 revolutions/day– n=5612.81065 degrees/day

• 1 day=107.088278 TU

– n=52.4129 degrees/TU• 1 radian=57.2957795 degrees

– n=.9147782342 radians/TU

• a=– a=1.061180 ER

• 1 ER=6378.1363 km

– a=6768.357 km

31

2

n

TLE Perturbations Effects

• The three perturbation effects in the TLE’s are mean motion rate, mean motion acceleration and B* a drag parameter

• The ballistic coefficient, BC, can be found from B*

TLE Identification Purposes

• The Satellite number

• The International Designation tells us the year of the satellite launch, launch number of year and section– For this satellite is 86017A, that means it was

the 17th launch of 1986 an it was the A section.

TLE Time

• The epoch is what time the values were recorded– The Time give was 93352.53502934

– This Translates to the 352nd day of 1993 which was December 18.

– To find the Hours, minutes and seconds just take the remainder divide by 24 to get the hours, take the remainder of that divide by 60 to get the minutes and take the remainder of that divide by 60 to get the seconds

– This should translate to 12 h. 50 min. and 26.535 sec.