orbital mechanics

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

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 Energykg in orbit

= −μ

rorbit+

= 1.9 × 106 Jkg

Kinetic Energykg in orbit

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

Total Energykg in orbit

= 32 × 106 Jkg

= 8.9kWhrs

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!

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)

WeightThrust

Drag, or

Total Energykg

=thrust × distance

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!

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!

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)

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

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

Energy in Orbit

<--Vis-Viva Equation

• Kinetic Energy

• Potential Energy

• Total Energy

Ekinetic ≡12

mv2 ⟹Ekinetic

Epotential ≡ −μmr

⟹Epotential

m= −

Etotal = constant =v2

= −μ2a

v2 = μ ( 2r

−1a )

Classical Parameters of Elliptical Orbits

The Classical Orbital Elements

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

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

• Parabolic escape orbit (a tends to infinity)

• Relationship between circular and parabolic orbits

vcircular =μr

vescape =2μr

vescape = 2vcircular

The Hohmann Transfer

vperigee

vapogee

First Maneuver Velocities• Initial vehicle velocity

• Needed final velocity

• Delta-V

v1 =μr1

vperigee =μr1

r1 + r2

Δv1 = vperigee − v1 =μr1

r1 + r2− 1

Second Maneuver Velocities• Initial vehicle velocity

• Delta-V

v2 =μr2

vapogee =μr2

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

Δvlow thrust = vc1 − vc2 =μr1

−μr2

Limitations on Launch Inclinations

Equator

Differences in Inclination

Line of Nodes

Choosing the Wrong Line of Apsides

Simple Plane Change

vperigee

v1vapogee

Optimal Plane Change

vperigee v1 vapogeev2

Δv2Δv1

First Maneuver with Plane Change Δi1

• Initial vehicle velocity

• Delta-V

v1 =μr1

vp =μr1

r1 + r2

Δv1 = v21 + v2

p − 2v1vp cos Δi1

Second Maneuver with Plane Change • Initial vehicle velocity

• Delta-V

v2 =μr2

va =μr2

r1 + r2

Δv2 = v22 + v2

a − 2v2va cos Δi2

Sample Plane Change Maneuver

Optimum initial plane change = 2.20°

Calculating Time in Orbit

Time in Orbit• Period of an orbit

• Mean motion (average angular velocity)

• Time since pericenter passage

➥M=mean anomaly

M ≡ nt = E − e sin E

n =μa3

P = 2πa3

Dealing with the Eccentric Anomaly• Relationship to orbit

• Relationship to true anomaly

• Calculating M from time interval: iterate

until it converges

Ei+1 = nt + e sin Ei

r = a(1 − e cos E)

tanθ2

=1 + e1 − e

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)

(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

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°

E = 2.317

r = a(1 − e cos E) = 12,387 km

tanθ2

=1 + e1 − e

⟹ θ = 160 deg

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

• Radial distance as function of orbital position

• Periapse and apoapse distances

• Angular momentum

p = a (1 − e2)

1 + e cos θ

rp = a(1 − e)

h = r × v

ra = a(1 + e)

h = μp

Velocity Components in Orbit

h = r × v

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

1 + e cos θ =pr

⇒ vr =r2 dθ

dte sin θ

Velocity Components in Orbit (cont.)

vr =r2 dθ

dte sin θ

he sin θp

pe sin θ

vr =μp

e sin θ

vθ = rdθdt

= rhr2

rvθ =

(1 + e cos θ)

tan γ =vr

e sin θ1 + e cos θ

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

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.

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”

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

= −μ2a

v = v2h +

Δv = v2h +

−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)

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

vpm = v2h +

rLLO= 0.8842 +

2(4667.9)1878

= 2.398kmsec

Δv = vpm − vcm = 2.398 −4667.91878

= 0.822kmsec

ΔV Requirements for Lunar MissionsTo:

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

LOI ΔV Based on Landing Site

LOI ΔV Including Loiter Effects

Lagrange Points

Non-Keplerian Orbits

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

Transfer to/from NRHO

Delta-V Costs for Lunar Orbits

Interplanetary Trajectory Types

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

C3 for Earth-Mars Transfer 1990-2045

Earth-Mars Transfer 2033

Earth-Mars Transfer 2037

Interplanetary Delta-V

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

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)

ΔvE =μE

rLEO−

ΔvM =μM

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)

Δvtotal = ΔvEarth escape + ΔvEarth−Mars + ΔvMars capture

ΔvMars capture =μMars

ΔvEarth escape =μEarth

ΔvEarth−Mars =μSun

rEarth orbit−

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

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

Clohessy-Wiltshire (“CW”) Equations 1

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

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

Clohessy-Wiltshire (“CW”) Equations 2

·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

“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

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

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

orbital mechanics - umd · 2020. 9. 10. · orbital mechanics enae 483/788d - principles of space...

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