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Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design

U N I V E R S I T Y O FMARYLAND

Orbital Mechanics

• Lecture #08 – September 22, 2016 • Planetary launch and entry overview • Energy and velocity in orbit • Elliptical orbit parameters • Orbital elements • Coplanar orbital transfers • Noncoplanar transfers • Time in orbit • Interplanetary trajectories • Low-thrust trajectories • Relative orbital motion (“proximity operations”)

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© 2016 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu

http://spacecraft.ssl.umd.edu



Space Launch - The Physics

• Minimum orbital altitude is ~200 km

• Circular orbital velocity there is 7784 m/sec

• Total energy per kg in orbit

Potential Energy

kg in orbit= � µ

rorbit+

µ

rE= 1.9⇥ 106 J

kg

Kinetic Energy

kg in orbit=

12

µ

r2orbit

= 30� 106 J

kg

Total Energy

kg in orbit= KE + PE = 32� 106 J

kg

2



Theoretical Cost to Orbit

• Convert to usual energy units

• Domestic energy costs are ~$0.05/kWhr

!Theoretical cost to orbit $0.44/kg

Total Energy

kg in orbit= 32� 106 J

kg= 8.888

kWhrs

kg

3



Actual Cost to Orbit

• SpaceX Falcon 9 – 13,150 kg to LEO – $65 M per flight – Lowest cost system currently

flying

• $4940/kg of payload • Factor of 11,000x higher

than theoretical energy costs!

4



What About Airplanes?

• For an aircraft in level flight,

• Energy = force x distance, so

• For an airliner (L/D=25) to equal orbital energy, d=81,000 km (2 roundtrips NY-Sydney)

5

Total Energykg

=thrust� distance

mass=

Td

m=

gd

L/D

WeightThrust

=LiftDrag

, ormg

T=

L

D



Equivalent Airline Costs?

• Average economy ticket NY-Sydney round-round-trip (Travelocity 9/3/09) ~$1300

• Average passenger (+ luggage) ~100 kg • Two round trips = $26/kg

– Factor of 60x more than electrical energy costs – Factor of 190x less than current launch costs

• But… you get to refuel at each stop!

6



Equivalence to Air Transport

• 81,000 km ~ twice around the world

• Voyager - one of only two aircraft to ever circle the world non-stop, non-refueled - once!

7



Orbital Entry - The Physics

• 32 MJ/kg dissipated by friction with atmosphere over ~8 min = 66kW/kg

• Pure graphite (carbon) high-temperature material: cp=709 J/kg°K

• Orbital energy would cause temperature gain of 45,000°K!

8



Newton’s Law of Gravitation

• Inverse square law

• Since it’s easier to remember one number,

• If you’re looking for local gravitational acceleration,

F =

GMm

r2

µ = GM

g =

µ

r2

9



Some Useful Constants

• Gravitation constant µ = GM – Earth: 398,604 km3/sec2

– Moon: 4667.9 km3/sec2

– Mars: 42,970 km3/sec2 – Sun: 1.327x1011 km3/sec2

• Planetary radii – rEarth = 6378 km

– rMoon = 1738 km

– rMars = 3393 km

10



Energy in Orbit

<--Vis-Viva Equation

K.E. =1

2mv

2=⇒

K.E.

m=

v2

2

P.E. = −µm

r=⇒

P.E.

m= −

µ

r

Constant =v2

2−

µ

r= −

µ

2a

• Kinetic Energy

• Potential Energy

• Total Energy

11

v2 = µ

�2r� 1

a

⇥



Classical Parameters of Elliptical Orbits

12



The Classical Orbital Elements

Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993

13



Implications of Vis-Viva

• Circular orbit (r=a)

• Parabolic escape orbit (a tends to infinity)

• Relationship between circular and parabolic orbits

vcircular =

!

µ

r

vescape =

!

2µ

r

vescape =√

2vcircular

14



The Hohmann Transfer

vperigee

v1

vapogee

v2r1

r2

15



First Maneuver Velocities

• Initial vehicle velocity

• Needed final velocity

• Delta-V

v1 =

!

µ

r1

vperigee =

!

µ

r1

!

2r2

r1 + r2

∆v1 =

!

µ

r1

"!

2r2

r1 + r2

− 1

#

16



Second Maneuver Velocities



• Delta-V

∆v2 =

!

µ

r2

"

1 −

!

2r1

r1 + r2

#

vapogee =

!

µ

r2

!

2r1

r1 + r2

v2 =

!

µ

r2

17



Implications of Hohmann Transfers

• Implicit assumption is made that velocity changes instantaneously - “impulsive thrust”

• Decent assumption if acceleration ≥ ~5 m/sec2 (0.5 gEarth)

• Lower accelerations result in altitude change during burn ⇒ lower efficiencies and higher ΔVs

• Worst case is continuous “infinitesimal” thrusting (e.g., ion engines) ⇒ ΔV between circular coplanar orbits r1 and r2 is

18

�VLow Thrust

= Vc1 � V

c2 =

rµ

r1�r

µ

r2



Limitations on Launch Inclinations

Equator

19



Differences in Inclination

Line of Nodes

20



Choosing the Wrong Line of Apsides

21



Simple Plane Change

vperigee

v1vapogee

v2

Δv2

22



Optimal Plane Change

vperigee v1 vapogee

v2

Δv2Δv1

23



First Maneuver with Plane Change Δi1



• Delta-V

v1 =�

µ

r1

vp =�

µ

r1

�2r2

r1 + r2

�v1 =�

v21 + v2

p � 2v1vp cos �i1

24



Second Maneuver with Plane Change Δi2



• Delta-V

�v2 =�

v22 + v2

a � 2v2va cos �i2

va =�

µ

r2

�2r1

r1 + r2

v2 =�

µ

r2

25



Sample Plane Change Maneuver

Optimum initial plane change = 2.20°

26



Calculating Time in Orbit

27



Time in Orbit

• Period of an orbit

• Mean motion (average angular velocity)

• Time since pericenter passage

➥M=mean anomaly

P = 2⇥

�a3

µ

n =�

µ

a3

M = nt = E � e sinE

28



Dealing with the Eccentric Anomaly

• Relationship to orbit

• Relationship to true anomaly

• Calculating M from time interval: iterate

until it converges

r = a (1� e cos E)

tan�

2=

�1 + e

1� etan

E

2

Ei+1 = nt + e sinEi

29



Example: Time in Orbit

• Hohmann transfer from LEO to GEO – h1=300 km --> r1=6378+300=6678 km

– r2=42240 km

• Time of transit (1/2 orbital period)

a =12

(r1 + r2) = 24, 459 km

ttransit =P

2= ⇥

�a3

µ= 19, 034 sec = 5h17m14s

30



Example: Time-based Position

Find the spacecraft position 3 hours after perigee

E=0; 1.783; 2.494; 2.222; 2.361; 2.294; 2.328; 2.311; 2.320; 2.316; 2.318; 2.317; 2.317; 2.317

Ej+1 = nt + e sin Ej = 1.783 + 0.7270 sin Ej

n =�

µ

a3= 1.650x10�4 rad

sec

e = 1� rp

a= 0.7270

31



Example: Time-based Position (cont.)

Have to be sure to get the position in the proper quadrant - since the time is less than 1/2 the period, the spacecraft has yet to reach apogee --> 0°<θ<180°

32

E = 2.317

tan�

2=

�1 + e

1� etan

E

2=⇥ � = 160 deg

r = a(1� e cosE) = 12, 387 km



Basic Orbital Parameters

• Semi-latus rectum (or parameter)

• Radial distance as function of orbital position

• Periapse and apoapse distances

• Angular momentum

h⃗ = r⃗ × v⃗

p = a(1 − e2)

r =p

1 + e cos θ

rp = a(1 − e) ra = a(1 + e)

h =

√

µp

33



Velocity Components in Orbit

34

r =p

1 + e cos �

vr =dr

dt=

d

dt

�p

1 + e cos �

⇥=�p(�e sin � d�

dt )(1 + e cos �)2

vr =pe sin �

(1 + e cos �)2d�

dt

1 + e cos � =p

r� vr =

r2 d�dt e sin �

p�⇤h = �⇤r ⇥�⇤v



Velocity Components in Orbit (cont.)

35

~h = ~r ⇥ ~v h = rv cos � = r

✓rd✓

dt

◆= r2

d✓

dt



Patched Conics• Simple approximation to multi-body motion (e.g.,

traveling from Earth orbit through solar orbit into Martian orbit)

• Treats multibody problem as “hand-offs” between gravitating bodies --> reduces analysis to sequential two-body problems

• Caveat Emptor: There are a number of formal methods to perform patched conic analysis. The approach presented here is a very simple, convenient, and not altogether accurate method for performing this calculation. Results will be accurate to a few percent, which is adequate at this level of design analysis.

36



Example: Lunar Orbit Insertion• v2 is velocity of moon

around Earth • Moon overtakes

spacecraft with velocity of (v2-vapogee)

• This is the velocity of the spacecraft relative to the moon while it is effectively “infinitely” far away (before lunar gravity accelerates it) = “hyperbolic excess velocity”

37



Planetary Approach Analysis

• Spacecraft has vh hyperbolic excess velocity, which fixes total energy of approach orbit

• Vis-viva provides velocity of approach

• Choose transfer orbit such that approach is tangent to desired final orbit at periapse

v =

!

v2

h+

2µ

r

∆v =

!

v2

h+

2µ

r−

!

µ

r

v2

2� µ

r= � µ

2a=

v2h

2

38



Patched Conic - Lunar Approach

• Lunar orbital velocity around the Earth

• Apogee velocity of Earth transfer orbit from initial 400 km low Earth orbit

• Velocity difference between spacecraft “infinitely” far away and moon (hyperbolic excess velocity)

vm =

!

µ

rm

=

!

398, 604

384, 400= 1.018

km

sec

va = vm

!

2r1

r1 + rm

= 1.018

!

6778

6778 + 384, 400= 0.134

km

sec

vh = vm − va = vm = 1.018 − 0.134 = 0.884km

sec

39



Patched Conic - Lunar Orbit Insertion

• The spacecraft is now in a hyperbolic orbit of the moon. The velocity it will have at the perilune point tangent to the desired 100 km low lunar orbit is

• The required delta-V to slow down into low lunar orbit is

40

vpm =

rv2h +

2µm

rLLO=

r0.8842 +

2(4667.9)

1878= 2.398

km

sec

�v = vpm � vcm = 2.398�r

4667.9

1878= 0.822

km

sec

�v = vpm � vcm = 2.398�r

4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec�v = vpm � vcm = 2.398�

r4667.9

1878= 0.822

km

sec



ΔV Requirements for Lunar Missions

To:

From:

Low EarthOrbit

LunarTransferOrbit

Low LunarOrbit

LunarDescentOrbit

LunarLanding

Low EarthOrbit

3.107km/sec

LunarTransferOrbit

3.107km/sec

0.837km/sec

3.140km/sec

Low LunarOrbit

0.837km/sec

0.022km/sec

LunarDescentOrbit

0.022km/sec

2.684km/sec

LunarLanding

2.890km/sec

2.312km/sec

41



LOI ΔV Based on Landing Site

42



LOI ΔV Including Loiter Effects

43



Interplanetary Trajectory Types

44



Interplanetary “Pork Chop” Plots• Summarize a number of

critical parameters – Date of departure – Date of arrival – Hyperbolic energy (“C3”) – Transfer geometry

• Launch vehicle determines available C3 based on window, payload mass

• Calculated using Lambert’s Theorem

45



C3 for Earth-Mars Transfer 1990-2045

46



Earth-Mars Transfer 2033

47



Earth-Mars Transfer 2037

48



Interplanetary Delta-V

49

C3 = V 2h

Hyperbolic excess velocity ⌘ Vh

Vreq =p

V 2esc + C3

�V =pV 2esc + C3 � Vc

2033 Window: �V = 3.55 km/sec

2037 Window: �V = 3.859 km/sec

�V in departure from 300 km LEO



Low-Thrust Trajectories

• Acceleration measured in milligees • Circular orbits remain circular • Vehicle spirals outbound over many orbits

• From any circular orbit, to escape is

50

�v = vc1 � vc2

�v

�v =

rµ

r



Low-Thrust ∆V to Moon

• From LEO to Moon’s orbit around Earth

• From Moon’s orbit to low lunar orbit (LLO)

51

�vE =

rµE

rLEO�r

µE

rMoon

�vM =

rµM

rLLO

�vtotal

= �vE

+�vM



Low-Thrust ∆V to Mars

52

�vEarth escape =

rµE

rLEO

�vEarth�Mars

=

rµsun

rEarth orbit

�r

µsun

rMars orbit

�vMars capture =

rµMars

rLMO

�vMars capture = �vEarth escape +�vEarth�Mars +�vMars capture



Low-Thrust Caveats

• This analysis assumes infinitesimal thrust • ∆V calculation should be conservative for any

higher thrust level • This will serve for preliminary analysis of low-

thrust mission requirements • More accurate ∆V (and time of flight) calculations

will depend on integrating the equations of motion and iterating to match boundary conditions

53



Hill’s Equations (Proximity Operations)

€

˙ ̇ x = 3n2x + 2n˙ y + adx

€

˙ ̇ y = −2n˙ x + ady

€

˙ ̇ z = −n 2z + adz

Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993

54



Clohessy-Wiltshire (“CW”) Equations

€

x(t) = 4 − 3cos(nt)[ ]xo +sin(nt)

n˙ x o +

2n

1− cos(nt)[ ] ˙ y o

€

y(t) = 6 sin(nt)− nt[ ]xo + yo −2n

1−cos(nt)[ ] ˙ x o +4sin(nt)− 3nt

n˙ y o

€

z( t) = zo cos(nt) +˙ z on

sin(nt)

€

˙ z ( t) = −zonsin(nt) + ˙ z o sin(nt)

55



“V-Bar” Approach

Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On-Orbit Satellite Servicing and Resupply, 15th USU Small Satellite Conference, 2001

56



“R-Bar” Approach

• Approach from along the radius vector (“R-bar”)

• Gravity gradients decelerate spacecraft approach velocity - low contamination approach

• Used for Mir, ISS docking approaches

Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On-Orbit Satellite Servicing and Resupply, 15th USU Small Satellite Conference, 2001

57



References for This Lecture

• Wernher von Braun, The Mars Project University of Illinois Press, 1962

• William Tyrrell Thomson, Introduction to Space Dynamics Dover Publications, 1986

• Francis J. Hale, Introduction to Space Flight Prentice-Hall, 1994

• William E. Wiesel, Spaceflight Dynamics MacGraw-Hill, 1997

• J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993

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