Orbital Mechanics - UMD · 2020. 9. 10. · Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Orbital Mechanics • Lecture #04

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 #04 – September 10, 2020 • 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 • Relative orbital motion (“proximity operations”) • Low-thrust trajectories

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© 2020 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

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Potential Energykg in orbit

= −μ

rorbit+

μrE

= 1.9 × 106 Jkg

Kinetic Energykg in orbit

=12

μr2orbit

= 30 × 106 Jkg

Total Energykg in orbit

= KE + PE = 32 × 106 Jkg



Theoretical Cost to Orbit• Convert to usual energy units

• Domestic energy costs are ~$0.11/kWhr

!Theoretical cost to orbit $1/kg

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Total Energykg in orbit

= 32 × 106 Jkg

= 8.9kWhrs

kg



Actual Cost to Orbit• SpaceX Falcon 9

– 22,800 kg to LEO – $62 M per flight – Lowest cost system currently

flying • $2720/kg of payload • Factor of 2700x higher than

theoretical energy costs!

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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)

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WeightThrust

=Lift

Drag, or

mgT

=LD

Total Energykg

=thrust × distance

mass=

Tdm

=gd

L/D



Equivalent Airline Costs?• Average economy ticket NY-Sydney round-trip

(Travelocity 5/28/19) ~$1000 • Average passenger (+ luggage) ~100 kg • Two round trips = $20/kg

– Factor of 20x more than electrical energy costs – Factor of 136x less than current launch costs

• But… you get to refuel at each stop!

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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!

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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! • Thus proving the comment about space travel,

“It’s utter bilge!” (Sir Richard Wooley, Astronomer Royal of Great Britain, 1956)

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Newton’s Law of Gravitation• Inverse square law

• Since it’s easier to remember one number,

• If you’re looking for local gravitational acceleration,

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F =GMm

r2

μ ≡ GM

g =μr2



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

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Energy in Orbit

<--Vis-Viva Equation

• Kinetic Energy

• Potential Energy

• Total Energy

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Ekinetic ≡12

mv2 ⟹Ekinetic

m=

12

v2

Epotential ≡ −μmr

⟹Epotential

m= −

μr

Etotal = constant =v2

2−

μr

= −μ2a

v2 = μ ( 2r

−1a )



Classical Parameters of Elliptical Orbits

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θ



The Classical Orbital Elements

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

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Implications of Vis-Viva• Circular orbit (r=a)

• Parabolic escape orbit (a tends to infinity)

• Relationship between circular and parabolic orbits

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vcircular =μr

vescape =2μr

vescape = 2vcircular



The Hohmann Transfer

vperigee

v1

vapogee

v2r1

r2

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First Maneuver Velocities• Initial vehicle velocity

• Needed final velocity

• Delta-V

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v1 =μr1

vperigee =μr1

2r2

r1 + r2

Δv1 = vperigee − v1 =μr1

2r2

r1 + r2− 1



Second Maneuver Velocities• Initial vehicle velocity


• Delta-V

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v2 =μr2

vapogee =μr2

2r1

r1 + r2

Δv2 = v2 − vapogee =μr2

1 −2r1

r1 + r2



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

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Δvlow thrust = vc1 − vc2 =μr1

−μr2



Limitations on Launch Inclinations

Equator

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Differences in Inclination

Line of Nodes

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Choosing the Wrong Line of Apsides

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Simple Plane Change

vperigee

v1vapogee

v2

Δv2

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Optimal Plane Change

vperigee v1 vapogeev2

Δv2Δv1

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First Maneuver with Plane Change Δi1

• Initial vehicle velocity


• Delta-V

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v1 =μr1

vp =μr1

2r2

r1 + r2

Δv1 = v21 + v2

p − 2v1vp cos Δi1



Second Maneuver with Plane Change • Initial vehicle velocity


• Delta-V

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v2 =μr2

va =μr2

2r1

r1 + r2

Δv2 = v22 + v2

a − 2v2va cos Δi2



Sample Plane Change Maneuver

Optimum initial plane change = 2.20°

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Calculating Time in Orbit

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Time in Orbit• Period of an orbit

• Mean motion (average angular velocity)

• Time since pericenter passage

➥M=mean anomaly

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M ≡ nt = E − e sin E

n =μa3

P = 2πa3

μ



Dealing with the Eccentric Anomaly• Relationship to orbit

• Relationship to true anomaly

• Calculating M from time interval: iterate

until it converges

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Ei+1 = nt + e sin Ei

r = a(1 − e cos E)

tanθ2

=1 + e1 − e

tanE2



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)

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a =12

(r1 + r2) = 24,459 km

ttransit =P2

= πa3

μ− 19,034 sec = 5h17m14s



Example: Time-based PositionFind 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

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

n =μa3

= 1.650 × 10−4 radsec

e = 1 −rp

a= 0.7270



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°

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E = 2.317

r = a(1 − e cos E) = 12,387 km

tanθ2

=1 + e1 − e

tanE2

⟹ θ = 160 deg



Basic Orbital Parameters• Semi-latus rectum (or parameter)

• Radial distance as function of orbital position

• Periapse and apoapse distances

• Angular momentum

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p = a (1 − e2)

r =p

1 + e cos θ

rp = a(1 − e)

h = r × v

ra = a(1 + e)

h = μp



Velocity Components in Orbit

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h = r × v

r =p

1 + e cos θ

vr =drdt

=ddt ( p

1 + e cos θ ) =−p (−e sin θ dθ

dt )(1 + e cos θ)2

vr =pe sin θ

(1 + e cos θ)2

dθdt

1 + e cos θ =pr

⇒ vr =r2 dθ

dte sin θ

p



Velocity Components in Orbit (cont.)

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vr =r2 dθ

dte sin θ

p=

he sin θp

=pμ

pe sin θ

vr =μp

e sin θ

vθ = rdθdt

= rhr2

=hr

=pμ

rvθ =

μp

(1 + e cos θ)

tan γ =vr

vθ=

e sin θ1 + e cos θ

h = r × v h = rv cos γ = r (rdθdt ) = r2 dθ

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.

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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”

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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

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v2

2−

μr

= −μ2a

=v2

h

2

v = v2h +

2μr

Δv = v2h +

2μr

−μr



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)

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vm =μrm

=398,604384,400

= 1.018kmsec

va = vm2r1

r1 + rm= 1.018

67786778 + 384,400

= 0.134kmsec

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



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

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vpm = v2h +

2μm

rLLO= 0.8842 +

2(4667.9)1878

= 2.398kmsec

Δv = vpm − vcm = 2.398 −4667.91878

= 0.822kmsec



ΔV Requirements for Lunar MissionsTo:

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

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LOI ΔV Based on Landing Site

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LOI ΔV Including Loiter Effects

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Lagrange Points

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Non-Keplerian Orbits

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Whitley and Martinez, “Options for Staging Orbits in Cis-lunar Space” IEEE Aerospace Conference, 2016



Transfer to/from NRHO

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Delta-V Costs for Lunar Orbits

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Interplanetary Trajectory Types

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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

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C3 for Earth-Mars Transfer 1990-2045

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Earth-Mars Transfer 2033

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Earth-Mars Transfer 2037

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Interplanetary Delta-V

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Δv for departure from 300 km orbit

Hyperbolic excess velocity ≡ Vh

C3 = V2h

Vreq = V2esc + C3

ΔV = V2esc + C3 − Vc

2033 window : ΔV = 3.55 km /sec

2037 window : ΔV = 3.859 km /sec



Low-Thrust Trajectories• Acceleration measured in milligees • Circular orbits remain circular • Vehicle spirals outbound over many orbits

• From any circular orbit, to escape is

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�v

Δv = vc1 − vc2

Δv =μr



Low-Thrust ∆V to Moon• From LEO to Moon’s orbit around Earth

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

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ΔvE =μE

rLEO−

μE

rMoon

ΔvM =μM

rLLO

Δvtotal = ΔvE + ΔvM



Low-Thrust ∆V to Mars• From LEO to Moon’s orbit around Earth

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

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Δvtotal = ΔvEarth escape + ΔvEarth−Mars + ΔvMars capture

ΔvMars capture =μMars

rLMO

ΔvEarth escape =μEarth

rLEO

ΔvEarth−Mars =μSun

rEarth orbit−

μSun

rMars orbit



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

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Hill’s Equations (Proximity Operations)

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

··z = − n2z + adz

··x = 3n2x + 2n ·y + adx

··y = − 2n ·x + ady

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Clohessy-Wiltshire (“CW”) Equations 1

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x(t) = [4 − 3 cos(nt)] x0 +sin(nt)

n·x0 +

2n [1 − cos(nt)] ·y0

y(t) = 6 [sin(nt) − nt] x0 −2n [1 − cos(nt)] ·x0 +

4 sin(nt) − 3ntn

·y0

z(t) = cos(nt)z0 +sin(nt)

n·z0



Clohessy-Wiltshire (“CW”) Equations 2

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·x(t) = 3n sin(nt)x0 + cos(nt) ·x0 + 2 sin(nt) ·y0

·y(t) = − 6n [1 − cos(nt)] x0 − 2 sin(nt) ·x0 + [4 cos(nt) − 3] ·y0

·z(t) = − n sin(nt)z0 + cos(nt) ·z0



“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

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“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

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Today’s Tools• Basic physics of orbital mechanics • DV requirements for simple orbit changes • Orbital maneuvering with plane changes • Time in orbits • Low-thrust trajectories • Interplanetary trajectories • Linearized relative orbital motion

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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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Orbital Mechanics - UMD · 2020. 9. 10. · Orbital Mechanics ENAE 483/788D - Principles of Space Systems Design U N I V E R S I T Y O F MARYLAND Orbital Mechanics • Lecture #04