Lecture06 Interplanetary 2011-2012 - ULiege · H. Curtis, Orbital Mechanics for Engineering...

Gaëtan Kerschen

Space Structures & Systems Lab (S3L)

6. Interplanetary Trajectories

Astrodynamics (AERO0024)

2

Course Outline

THEMATIC UNIT 1: ORBITAL DYNAMICS

Lecture 02: The Two-Body Problem

Lecture 03: The Orbit in Space and Time

Lecture 04: Non-Keplerian Motion

THEMATIC UNIT 2: ORBIT CONTROL

Lecture 05: Orbital Maneuvers

Lecture 06: Interplanetary Trajectories

3

Motivation

4


6.1 Patched conic method

6.2 Lambert’s problem

6.3 Gravity assist

5


6.1 Patched conic method

6.1.1 Problem statement

6.1.2 Sphere of influence

6.1.3 Planetary departure

6.1.4 Hohmann transfer

6.1.5 Planetary arrival

6.1.6 Sensitivity analysis and launch windows

6.1.7 Examples

6

?

Hint #1: design the Earth-Mars transfer using known concepts

Hint #2: division into simpler problems

Hint #3: patched conic method

7

What Transfer Orbit ? Constraints ?

Dep.

Dep.

Dep.

Arr.

Arr.

Arr.

“∆∆∆∆V1”

“∆∆∆∆V2”

Motion in the heliocentric

reference frame

8

Planetary Departure ? Constraints ?

?

Motion in the planetary

reference frame

β

∆∆∆∆V1

v∞/Earth Sunv

9

Planetary Arrival ? Similar Reasoning

Departure hyperbola

Arrival hyperbola

Transfer ellipse

SOISOI

10

Patched Conic Method

Three conics to patch:

1. Outbound hyperbola (departure)

2. The Hohmann transfer ellipse (interplanetary travel)

3. Inbound hyperbola (arrival)


11


Approximate method that analyzes a mission as a sequence of 2-body problems, with one body always being the spacecraft.

If the spacecraft is close enough to one celestial body, the gravitational forces due to other planets can be neglected.

The region inside of which the approximation is valid is called the sphere of influence (SOI) of the celestial body. If the spacecraft is not inside the SOI of a planet, it is considered to be in orbit around the sun.


12



Very useful for preliminary mission design (delta-v requirements and flight times).

But actual mission design and execution employ the most accurate possible numerical integration techniques.

13

Sphere of Influence (SOI) ?

Let’s assume that a spacecraft is within the Earth’s SOI if the gravitational force due to Earth is larger than the gravitational force due to the sun.

2 2, ,

E sat S sat

E sat S sat

Gm m Gm m

r r>

5, 2.5 10 kmE satr < ×

The moon lies outside the SOI and is in orbit about the sun like an asteroid !


14

ρ

d

1m

2mjm

r

Central body: sun or planet

Spacecraft

Third body: sun or planet

O

3 3 3jGmr d

µρ

+ = − +

ρdr r&&

Disturbing function (L04)

Sphere of Influence (SOI)


15

If the Spacecraft Orbits the Planet

( )3 3 3

p v spsvpv pv s

pv sv sp

G m mGm

r r r

+ + = − +

rrr r&&

p:planet v: vehicle

s:sun

pv p s− =r A P&&

Primary gravitational

acceleration due to the planet

Perturbation acceleration due

to the sun


16

If the Spacecraft Orbits the Sun

( )3 3 3

pv sps vsv sv p

sv pv sp

G m mGm

r r r

++ = − +

r rr r&&

p:planet v: vehicle

s:sun

sv s p− =r A P&&

Primary gravitational

acceleration due to the sun

Perturbation acceleration due

to the planet


17

SOI: Correct Definition due to Laplace

It is the surface along which the magnitudes of the acceleration satisfy:

p s

s p

P P

A A=

Measure of the deviation of the vehicle’s orbit from the Keplerian orbit arising from the planet acting by itself

Measure of the planet’s influence on the orbit of the vehicle relative to the sun

2

5p

SOI sps

mr r

m

≈


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SOI: Correct Definition due to Laplace

p s

s p

P P

A A>If the spacecraft is inside the SOI of the planet.

The previous (incorrect) definition was 1p

s

A

A>


19

SOI Radii

Planet

Mercury 1.13x105 45

Venus 6.17x105 100

Earth 9.24x105 145

Mars 5.74x105 170

Jupiter 4.83x107 677

Neptune 8.67x107 3886

SOI Radius (km)SOI radius (body radii)

OK !


20

Validity of the Patched Conic Method

The Earth’s SOI is 145 Earth radii.

This is extremely large compared to the size of the Earth:

The velocity relative to the planet on an escape hyperbola is considered to be the hyperbolic excess velocity vector.

This is extremely small with respect to 1AU:

During the elliptic transfer, the spacecraft is considered to be under the influence of the Sun’s gravity only. In other words, it follows an unperturbed Keplerian orbit around the Sun.


SOIv v∞≈

21

Outbound Hyperbola

2 2 2 2 2esc SOI escv v v v v∞= + ≈ +

The spacecraft necessarily escapes using a hyperbolic trajectory relative to the planet.

Is vSOI the velocity on the transfer orbit ?

Lecture 02:

Hyperbolic excess speed

When this velocity vector is added to the planet’s heliocentric velocity, the result is the spacecraft’s heliocentric velocity on the interplanetary elliptic transfer orbit at the SOI in the solar system.


22

Magnitude of V SOI

The velocity vD of the spacecraft relative to the sun is imposed by the Hohmann transfer (i.e., velocity on the transfer orbit).

H. Curtis, Orbital Mechanics for Engineering Students, Elsevier.


23

Magnitude of V SOI

By subtracting the known value of the velocity v1 of the planet relative to the sun, one obtains the hyperbolic excess speed on the Earth escape hyperbola.

( )2

11 1 2

21sun

SOI D

Rv v v v

R R R

µ∞

= − = − ≈

+

Lecture05Imposed

Known


24

Direction of V SOI

What should be the direction of vSOI ?

For a Hohmann transfer, it should be parallel to v1.


25

Parking Orbit

A spacecraft is ordinary launched into an interplanetary trajectory from a circular parking orbit. Its radius equals the periapse radius rp of the departure hyperbola.



26

∆V Magnitude and Location

2

2

1

1

ha

eµ=

−

2 1eh

v

µ∞

−= 2 2p

p

h r vr

µ∞= +

2

1 pr ve

µ∞= +

p cp p

hv v v

r r

µ∆ = − = − 1 1cos

eβ −=

2

(1 )p

hr

eµ=

+Lecture 02:

va

µ∞ =

knownknown


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Planetary Departure: Graphically


Departure to outer or inner planet ?


28

Circular, Coplanar Orbits for Most Planets

Planet

Mercury 7.00º 0.206

Venus 3.39º 0.007

Earth 0.00º 0.017

Mars 1.85º 0.094

Jupiter 1.30º 0.049

Saturn 2.48º 0.056

Uranus 0.77º 0.046

Neptune 1.77º 0.011

Pluto 17.16º 0.244

Inclination of the orbit to the ecliptic plane

Eccentricity


29

Governing Equations


( )

( )

21

1 1 2

12

2 1 2

21

21

sunD

sunA

Rv v

R R R

Rv v

R R R

µ

µ

− = −

+

− = −

+

v2 - vA, vD - v1 >0 for transfer to an outer planet

v2 - vA, vD - v1 <0 for transfer to an inner planet

Signs ?

vDv1

vAv2

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Schematically


Transfer to inner planet

Transfer to outer planet

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For an outer planet, the spacecraft’s heliocentric approach velocity vA is smaller in magnitude than that of the planet v2.

Arrival at an Outer Planet


2 A∞+ =v v v

2 A>v v

and have opposite signs.

v∞2v

32

Arrival at an Outer Planet


The spacecraft crosses the forward portion of the SOI

33

If the intent is to go into orbit around the planet, then ∆must be chosen so that the ∆v burn at periapse will occur at the correct altitude above the planet.

Enter into an Elliptic Orbit

2

21p

p

rr v

µ∞

∆ = +

2, ,

(1 ) 2 (1 )p hyp p capture

p p p p

h e ev v v v

r r r r

µ µ µ∞

+ +∆ = − = − = + −


34

Planetary Flyby

Otherwise, the specacraft will simply continue past periapse on a flyby trajectory exiting the SOI with the same relative speed v∞ it entered but with the velocity vector rotated through the turn angle δ.

1 12sin

eδ −=

2

1 pr ve

µ∞= +


35

Sensitivity Analysis: Departure

The maneuver occurs well within the SOI, which is just a point on the scale of the solar system.

One may therefore ask what effects small errors in position and velocity (rp and vp) at the maneuver point have on the trajectory (target radius R2 of the heliocentric Hohmann transfer ellipse).

1

2 12

12

2

2

12

p p p

D D p p D p

sun

vr r vR

R vR v v r r v v

µδ δδ µ

µ

∞

∞

+ = + −


36

Sensitivity Analysis: Earth -Mars, 300km Orbit

11 3 2 3 21

6 61 2

1.327 10 km / , 398600 km /

149.6 10 km, 227.9 10 km, 6678 km

32.73 km / , 2.943 km /

sun

p

D

s s

R R r

v s v s

µ µ

∞

= × =

= × = × =

= =

2

2

3.127 6.708p p

p p

r vR

R r v

δ δδ = +

A 0.01% variation in the burnout speed vp changes the target radius by 0.067% or 153000 km.

A 0.01% variation in burnout radius rp (670 m !) produces an error over 70000 km.


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Sensitivity Analysis: Launch Errors

Ariane V

Trajectory correction maneuvers are clearly mandatory.



Sensitivity Analysis: Arrival

The heliocentric velocity of Mars in its orbit is roughly 24km/s.

If an orbit injection were planned to occur at a 500 km periapsis height, a spacecraft arriving even 10s late at Mars would likely enter the atmosphere.

Cassini-Huygens

40

Existence of Launch Windows

Phasing maneuvers are not practical due to the large periods of the heliocentric orbits.

The planet should arrive at the apse line of the transfer ellipse at the same time the spacecraft does.


41

Rendez-vous Opportunities

1 10 1

2 20 2

2 1

n t

n t

θ θθ θφ θ θ

= += +

= −( )0 2 1n n tφ φ= + −

( )0 0 2 12 synn n Tφ π φ− = + −1 2

2synT

n n

π=−

Synodic period

1 2

1 2syn

TTT

T T=

−


42

Transfer Time

3/ 2

1 212 2

sun

R Rt

πµ

+ =

0 2 12n tφ π= −

Lecture 02


43

Earth -Mars Example

365.26 687.99777.9 days

365.26 687.99synT×= =−

It takes 2.13 years for a given configuration of Mars relative to the Earth to occur again.

712

0

2.2362 10 258.8 days

44

t s

φ= × =

= o

The total time for a manned Mars mission is

258.8 + 453.8 + 258.8 = 971.4 days = 2.66 years


446.1.7 Examples

Earth -Jupiter Example: Hohmann

Galileo’s original mission was designed to use a direct Hohmann transfer, but following the loss of Challenger Galileo's intended Centaur booster rocket was no longer allowed to fly on Shuttles. Using a less-powerful solid booster rocket instead, Galileo used gravity assists instead.

45

Earth -Jupiter Example: Hohmann

( )

( )

21

1 1 2

12

2 1 2

Velocity when leaving Earth's SOI:

21 8.792km/s

Velocity relative to Jupiter at Jupiter's SOI:

21 5.643km/s

Transfer time: 2.732 years

E sunD

J sunA

Rv v v

R R R

Rv v v

R R R

µ

µ

∞

∞

− = = − =

+

− = = − =

+

6.1.7 Examples

46

Earth -Jupiter Example: Departure

2

2

Velocity on a circular parking orbit (300km):

7.726km/s

27.726km/s 6.298km/s

1 2.295

Ec

E

p

p

vR h

v vr

r ve

µ

µ

µ

∞

∞

= =+

∆ = + − =

= + =

6.1.7 Examples

47

Earth -Jupiter Example: Arrival

J

2

Final orbit is circular with radius=6R

2 (1 )24.95 17.18 7.77km/s

e=1.108

p p

ev v

r r

µ µ∞

+∆ = + − = − =

6.1.7 Examples

48

Hohmann Transfer: Other Planets

6.1.7 Examples

Planet

Mercury

Venus

Mars

Jupiter

Saturn

Pluto

v∞ departure (km/s)

Transfer time (days)

7.5

2.5

2.9

8.8

10.3

11.8

105

146

259

998

2222

16482

Assumption of circular, co-planar orbits and tangential burns

49

Venus Express: A Hohmann -Like Transfer

6.1.7 Examples

C3 = 7.8 km2/s2

Time: 154 days

C3 = 6.25 km2/s2

Time: 146 days

Real data

Hohmann

Why ?

6.1.7 Examples

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Motivation

Section 6.1 discussed Hohmann interplanetary transfers, which are optimal with respect to fuel consumption.

Why should we consider nontangential burns (i.e., non-Hohmann transfer) ?


L05

53

Non-Hohmann Trajectories


Solution using Lambert’s theorem (Lecture 05):

If two position vectors and the time of flight are known, then the orbit can be fully determined.

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Venus Express Example


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Porkchop Plot: Visual Design Tool


Departure date

Arrival date

C3 contours

C3 contours

In porkchop plots, orbits are considered to be non-coplanar and elliptic.

Type I transfer for piloted: thespacecraft travels less than a 180°true anomaly

Type II transfer for cargo: thespacecraft travels more than a 180°true anomaly

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6.3 Gravity assist

6.3.1 Basic principle

6.3.2 Real-life examples

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∆V Budget: Earth Departure

Planet

Mercury

Venus

Mars

Jupiter

Saturn

Pluto

C3(km2/s2)

[56.25]

6.25

8.41

77.44

106.09

[139.24]

Assumption of circular, co-planar orbits and

tangential burns

SOYUZ

63

∆V Budget: Arrival at the Planet

A spacecraft traveling to an inner planet is accelerated by the Sun's gravity to a speed notably greater than the orbital speed of that destination planet.

If the spacecraft is to be inserted into orbit about that inner planet, then there must be a mechanism to slow the spacecraft.

Likewise, a spacecraft traveling to an outer planet is decelerated by the Sun's gravity to a speed far less than the orbital speed of that outer planet. Thus there must be a mechanism to accelerate the spacecraft.


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Prohibitive ∆V Budget ? Use Gravity Assist

Also known as planetary flyby trajectory, slingshot maneuver and swingby trajectory.

Useful in interplanetary missions to obtain a velocity change without expending propellant.

This free velocity change is provided by the gravitational field of the flyby planet and can be used to lower the ∆v cost of a mission.


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What Do We Gain ?

Spacecraft inbound velocity

Spacecraft outbound velocity

Vout = Vin

SOI


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Gravity Assist in the Heliocentric Frame

SOI

Planet’s sun relative velocity

Resultant Vin

Resultant Vout

, ,out in∞ ∞= −∆v v v


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A Gravity Assist Looks Like an Elastic Collision

Inertial frame

Frame attached to the train

Inertial frame

Frame attached to the train


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Leading -Side Planetary Flyby


A leading-side flyby results in a decrease in the spacecraft’s heliocentric speed (e.g., Mariner 10 and Messenger).

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Trailing -Side Planetary Flyby


A trailing-side flyby results in an increase in the spacecraft’s heliocentric speed (e.g., Voyager and Cassini-Huygens).

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What Are the Limitations ?

Launch windows may be rare (e.g., Voyager).

Presence of an atmosphere (the closer the spacecraft can get, the more boost it gets).

Encounter different planets with different (possibly harsh) environments.

What about flight time ?


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VVEJGA

See Lecture 1

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Messenger


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Hohmann Transfer vs. Gravity Assist

PlanetTransfer time

(days)

Mercury 2400

Saturn 2500

Remark: the comparison between the transfer times is difficult, because it depends on the target orbit. The transfer time for gravity assist mission is the time elapsed between departure at the Earth and first arrival at the planet.

C3 (km2/s2)

[56.25]

106.09

Transfer time (days)

105

C3 (km2/s2)

16.4

16.6

Real mission

Messenger

Cassini Huygens

2222

Gravity assist


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New Horizons: Mission to Pluto


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Direct Transfer Was Also Envisioned


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Launch Window Combining GA and Direct


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6.1 PATCHED CONIC METHOD







6.1.7 Examples

6.2 LAMBERT’S PROBLEM

6.3 GRAVITY ASSIST



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Even More Complex Trajectories…

Gaëtan Kerschen

Space Structures & Systems Lab (S3L)


Astrodynamics (AERO0024)

Lecture06 Interplanetary 2011-2012 - ULiege · H. Curtis, Orbital Mechanics for Engineering...

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