Astrodynamics

ReferencesP&M Ch 3SMAD Chs 6 & 7G&F Ch 4BMW Vallado

ReferencesP&M Ch 3SMAD Chs 6 & 7G&F Ch 4BMW Vallado

Brief History of Orbital MechanicsBasic Facts about OrbitsOrbital ElementsPerturbationsSatellite Coverage

Brief History of Orbital MechanicsBasic Facts about OrbitsOrbital ElementsPerturbationsSatellite Coverage

A Brief History of Orbital Mechanics

Aristotle (384-322 BC)Ptolemy (87-150 AD)Nicolaus Copernicus (1473-1543)Tycho Brahe (1546-1601)Johannes Kepler (1571-1630)Galileo Galilei (1564-1642)Sir Isaac Newton (1643-1727)

Aristotle (384-322 BC)Ptolemy (87-150 AD)Nicolaus Copernicus (1473-1543)Tycho Brahe (1546-1601)Johannes Kepler (1571-1630)Galileo Galilei (1564-1642)Sir Isaac Newton (1643-1727)

Kepler’s Laws

I. The orbit of each planet is an ellipse with the Sun at one focus.

II. The line joining the planet to the Sun sweeps out equal areas in equal times.

III. The square of the period of a planet is proportional to the cube of its mean distance to the sun.

I. The orbit of each planet is an ellipse with the Sun at one focus.

II. The line joining the planet to the Sun sweeps out equal areas in equal times.

III. The square of the period of a planet is proportional to the cube of its mean distance to the sun.

Kepler’s First Two Laws

I. The orbit of each planet is an ellipse with the Sun at one focus.I. The orbit of each planet is an ellipse with the Sun at one focus.

II. The line joining the planet to the Sun sweeps out equal areas in equal times.II. The line joining the planet to the Sun sweeps out equal areas in equal times.

apoapsis periapsis

Kepler’s Third Law

III. The square of the period of a planet is proportional to the cube of its mean distance to the sun.

III. The square of the period of a planet is proportional to the cube of its mean distance to the sun.

µπ

3

2 aT =

Here T is the period, a is the semimajor axis of the ellipse, and m is the gravitational parameter (depends on mass of central body)

Here T is the period, a is the semimajor axis of the ellipse, and m is the gravitational parameter (depends on mass of central body)

2311sunsun

235

skm 1032715.1

skm 1098601.3−

−⊕⊕

×==

×==

GM

GM

µ

µ

Mean Motion

• The Mean Motion is defined as

• The period can be written in terms of the mean motion as

• The Mean Anomaly is defined as

where tp is the time of periapsis passage

• The Mean Motion is defined as

• The period can be written in terms of the mean motion as

• The Mean Anomaly is defined as

where tp is the time of periapsis passage

3an µ

=

nT /2π=

)( pttnM −=

Earth Satellite Orbit Periods

Orbit Altitude (km) Period (min)LEO 300 90.52LEO 400 92.56MEO 3000 150.64GPS 20232 720GEO 35786 1436.07

Orbit Altitude (km) Period (min)LEO 300 90.52LEO 400 92.56MEO 3000 150.64GPS 20232 720GEO 35786 1436.07

Newton’s Laws

• Kepler’s Laws were based on observation data: “curve fits”

• Newton established the theory Universal Gravitational Law

Second Law

• Kepler’s Laws were based on observation data: “curve fits”

• Newton established the theory Universal Gravitational Law

Second Law

2rGMmFg −=

M

mr

rF &&rr

m=

21311

skgm 10672.6

−−−

×=G

Universal Gravitational Constant

Elliptical Orbits

• Planets, comets, and asteroids orbit the Sun in ellipses

• Moons orbit the planets in ellipses• Artificial satellites orbit the Earth in ellipses• To understand orbits, you need to

understand ellipses (and other conic sections)

• But first, let’s study circular orbits: A circle is a special case of an ellipse

• Planets, comets, and asteroids orbit the Sun in ellipses

• Moons orbit the planets in ellipses• Artificial satellites orbit the Earth in ellipses• To understand orbits, you need to

understand ellipses (and other conic sections)

• But first, let’s study circular orbits: A circle is a special case of an ellipse

Circular Orbits

• Speed of satellite in circular orbit depends on radius

• If an orbiting object at a particular radius has a speed < vc, then it is in an elliptical orbit with lower energy

• If an orbiting object at radius r has a speed > vc, then it is in a higher-energy orbit: elliptical, parabolic, or hyperbolic

• Speed of satellite in circular orbit depends on radius

• If an orbiting object at a particular radius has a speed < vc, then it is in an elliptical orbit with lower energy

• If an orbiting object at radius r has a speed > vc, then it is in a higher-energy orbit: elliptical, parabolic, or hyperbolic

rvc

µ=

a=r

v

The Energy of an Orbit• Orbital energy is the sum of the kinetic

energy, mv2/2, and the potential energy, -µm/r• Customarily, we use the specific mechanical

energy, E (i.e., the energy per unit mass of satellite)

• From this definition of energy, we can develop the following facts

E<0 ⇔ orbit is elliptical or circularE=0 ⇔ orbit is parabolicE>0 ⇔ orbit is hyperbolic

• Orbital energy is the sum of the kinetic energy, mv2/2, and the potential energy, -µm/r

• Customarily, we use the specific mechanical energy, E (i.e., the energy per unit mass of satellite)

• From this definition of energy, we can develop the following facts

E<0 ⇔ orbit is elliptical or circularE=0 ⇔ orbit is parabolicE>0 ⇔ orbit is hyperbolic

aE

rvE

2

2

2 µµ −=⇔−=

Properties of Ellipses

abr

p=a(1-e2)

ae

2a-p

focusvacantfocus

ν

aa(1+e) a(1-e)

Facts About Elliptical Orbits• Periapsis is the closest point of the orbit to the

central bodyrp = a(1-e)

• Apoapsis is the farthest point of the orbit from the central body

ra = a(1+e)

• Velocity at any point isv = (2E+2µ/r)1/2

• Escape velocity at any point isv = (2µ/r)1/2

• Periapsis is the closest point of the orbit to the central body

rp = a(1-e)• Apoapsis is the farthest point of the orbit from

the central bodyra = a(1+e)

• Velocity at any point isv = (2E+2µ/r)1/2

• Escape velocity at any point isv = (2µ/r)1/2

Orbital Elements

ων

iΩI

J

K

n

Equatorial plane

Orbital plane

Orbit is defined by 6 orbital elements (oe’s): semimajor axis, a; eccentricity, e; inclination, i; right ascension of ascending node, Ω; argument of periapsis, ω; and true anomaly, ν

Orbital Elements(continued)

• Semimajor axis a determines the size of the ellipse• Eccentricity e determines the shape of the ellipse• Two-body problem

a, e, i, Ω, and ω are constant6th orbital element is the angular measure of satellite motion in the orbit – 2 angles are commonly used:

• True anomaly, ν• Mean anomaly, M

• In reality, these elements are subject to various perturbations

Earth oblateness (J2)atmospheric dragsolar radiation pressuregravitational attraction of other bodies

Instantaneous Access Area

Example: Space shuttleExample: Space shuttleHR

RK

RK

KIAA

e

e

A

eA

A

+=

×=

=

−=

λ

π

λ

cos

1055604187.2

2

)cos1(

28

2

km

2km 628,476,1124.179551.0cos

km 300Hkm 6378

=

=⇒=

==

IAA

Re

oλλ

Re

H

λ

IAA

Elevation AngleElevation angle, ε, is measured up from horizon to target

Minimum elevation angle is typically based on the performance of an antenna or sensor

IAA is determined by same formula, but the Earth central angle, λ, is determined from the geometry shown

Elevation angle, ε, is measured up from horizon to target

Minimum elevation angle is typically based on the performance of an antenna or sensor

IAA is determined by same formula, but the Earth central angle, λ, is determined from the geometry shown

The angle η is called the nadir angleThe angle ρ is called the apparent Earth radius

The angle η is called the nadir angleThe angle ρ is called the apparent Earth radius

R⊕

λλ0 ε

η ρD

Geometry of Earth-Viewing

R⊕

λλ0 ε

η ρD

• Given altitude H, we can statesin ρ = cos λ0 = R⊕ / (R⊕+H)ρ+ λ0 = 90°

• For a target with known position vector, λ is easily computedcos λ = cos δs cos δt cos ∆L + sin δs sin δt

• Then tan η = sin ρ sin λ / (1- sin ρ cos λ)• And η + λ + ε = 90° and D = R⊕ sin λ / sin η

• Given altitude H, we can statesin ρ = cos λ0 = R⊕ / (R⊕+H)ρ+ λ0 = 90°

• For a target with known position vector, λ is easily computedcos λ = cos δs cos δt cos ∆L + sin δs sin δt

• Then tan η = sin ρ sin λ / (1- sin ρ cos λ)• And η + λ + ε = 90° and D = R⊕ sin λ / sin η

Algorithm for SSP, Ground Track

• Compute position vector in ECI• Determine Greenwich Sidereal Time θg at epoch, θg0

• Latitude is δs = sin-1(r3/r)• Longitude is Ls = tan-1(r2/r1)- θg0

• Propagate position vector in “the usual way”• Propagate GST using θg = θg0 +ω⊕(t-t0)

where ω⊕ is the angular velocity of the Earth• Notes:

http://www.aoe.vt.edu/~chall/courses/aoe4134/sidereal.pdfhttp://aa.usno.navy.mil/data/docs/WebMICA_2.htmlhttp://tycho.usno.navy.mil/sidereal.html

• Compute position vector in ECI• Determine Greenwich Sidereal Time θg at epoch, θg0

• Latitude is δs = sin-1(r3/r)• Longitude is Ls = tan-1(r2/r1)- θg0

• Propagate position vector in “the usual way”• Propagate GST using θg = θg0 +ω⊕(t-t0)

where ω⊕ is the angular velocity of the Earth• Notes:

http://www.aoe.vt.edu/~chall/courses/aoe4134/sidereal.pdfhttp://aa.usno.navy.mil/data/docs/WebMICA_2.htmlhttp://tycho.usno.navy.mil/sidereal.html

Ground TrackThis plot is for a satellite in a nearly circular orbit

0 60 120 180 240 300 360−90

−60

−30

0

30

60

90

ISS (ZARYA)

longitude

latit

ude

Ground TrackThis plot is for a satellite in a highly elliptical orbit

0 60 120 180 240 300 360−90

−60

−30

0

30

60

901997065B

longitude

latit

ude

Error Sources

Error Budgets

Hohmann Transfer•Initial LEO orbit has radius r1,

velocity vc1

•Desired GEO orbit has radius r2, velocity vc2

•Impulsive ∆v is applied to get on geostationary transfer orbit (GTO) at perigee

•Coast to apogee and apply another impulsive ∆v

•Initial LEO orbit has radius r1, velocity vc1

•Desired GEO orbit has radius r2, velocity vc2

•Impulsive ∆v is applied to get on geostationary transfer orbit (GTO) at perigee

•Coast to apogee and apply another impulsive ∆v

r1

r2

vc1 ∆v1v1

∆v2v2

vc2

1211

221 rrrrv µµµ −−=∆ +

2122

222 rrrrv +−−=∆ µµµ

LEO

GEO

GTO

Earth Oblateness Perturbations

• Earth is non-spherical, and to first approximation is an oblate spheroid

• The primary effects are on Ω and ω:

• Earth is non-spherical, and to first approximation is an oblate spheroid

• The primary effects are on Ω and ω:

)sin54()1(4

3

cos)1(2

3

2222

22

222

22

iea

nRJ

iea

nRJ

e

e

−−

−=

−=Ω

ω&

&

M

mr

3

2 100826.1

−

×−=J

The Oblateness Coefficient

Main Applications of J2 Effects

• Sun-synchronous orbits:The rate of change of Ω can be chosen so that the orbital plane maintains the same orientation with respect to the sun throughout the year

• Critical inclination orbits:The rate of change of ω can be made zero by selecting i ≈ 63.4°

• Sun-synchronous orbits:The rate of change of Ω can be chosen so that the orbital plane maintains the same orientation with respect to the sun throughout the year

• Critical inclination orbits:The rate of change of ω can be made zero by selecting i ≈ 63.4°