Today in Astronomy 111: introduction to three-body systemsdmw/ast111/Lectures/Lect_11b.pdfToday in Astronomy 111: multiple- body systems Two simple topics in systems of three bodies:

6 October 2011 Astronomy 111, Fall 2011 1

Today in Astronomy 111: multiple-body systems

Two simple topics in systems of three bodies: Transfer orbits, and how

asteroids get knocked from orbit to orbit I

The two-body potential without Coriolis force: Lagrange points

The path the MESSENGER probe took to Mercury, involving six gravity boosts and five burns (JHU-APL/NASA).

http://messenger.jhuapl.edu/index.php�http://saturn1.jpl.nasa.gov/home/index.cfm�


Hohmann transfer orbits

How do particles change orbits? By suffering collisions that result in the exchange of

energy and momentum by the colliders. (Elastic collisions, that is; no contact except with each other’s gravitational forces.)

We usually do this on purpose in reverse: endow particles (spacecraft) with the ability to change their own energy and momentum (rockets).

Animation by Shane Ross (Caltech)

http://www.cds.caltech.edu/~shane/movies/�


Hohmann transfer orbits (continued)

Basic scheme: change between two orbits by entering and leaving a third orbit that intersects the other two, changing energy and momentum at the intersections. All one needs to specify such orbits is the vis viva

equation,

which we can write once for each orbit at the intersection, and from the difference in calculate the energy required. (From direction change, calculatemomentum required.)

2 2 1 ,v GMr a

= −

2 ,vM

m



The most foolproof way to do this is to use the Hohmann transfer orbit:• Consider the simple case of two circular orbits, radii

and a transfer orbit tangent to the smaller orbit at periapse, and to the larger orbit at apoapse:

• Suppose that the velocity is changed instantaneously at transfer-orbit periapse andapoapse, by brief application of large thrust.

1 2 and ,r r

1r

2r

2a( ) 1 21 2

1 or ;2 .f a r ra r r

ε= ± == +

M

mPeriapse

Apoapse

Terminology

The Greek prefixes peri- and apo- mean close to (literally, “around”) and far from, respectively. The Greek suffixes helion, geon, asteron, apse refer respectively (in accusative form) to the Sun, Earth, star, and center. So we will use the wordsPerihelion/Aphelion: closest/furthest point from the Sun in the orbit of a planet or planetesimal in the Solar system. Perigee/Apogee: closest/furthest point from the Earth in the orbit of an Earth satellite (Moon or human-made).Periastron/Apoastron: closest/furthest point from a star in the orbit of a companion star or planet. Periapse/Apoapse: closest/furthest point from the focus containing the center of mass to any orbit about that focus. Note that the focus is not in general the same as the center! 6 October 2011 Astronomy 111, Fall 2011 5



• The advantage of this configuration is that the orbital velocities of transfer orbit and initial/final orbit are parallel. Indeed, as we’ll see, the thrust has to be applied to speed up the spacecraft at both periapse and apoapse.

• From vis viva, 2 2

,1 1 1 1

2 2,

2 2

, ,

2 1 2 1, ,

2 1 , .

Note that and .

i t p

t a f

f t a t p i

GMv GM v GMr r r r a

GMv GM vr a r

v v v v

= − = = −

= − =

> >



• So thrust has to be applied in such a way as to increase the spacecraft’s momentum without changing direction, both to leave the initial orbit and to circularize in the final one. The changes in momentum at periapse and apoapse are

( )

( )

1,

1

2

1 1 2

1,

2 1 2

2 1

2 1 and, similarly,

21 .

p t p i

a f t a

rGMp m v v mr a

rGMmr r r

rGMp m v v mr r r

∆ = − = − −

= − +

∆ = − = − +



• Supposing the spacecraft is capable of a large thrust (force) F, the duration of the burns at periapse and apoapse are given by

• The time it takes to reach its final destination is half the orbital period of the transfer orbit:

:F p t= ∆ ∆

2

1 1 2

1

2 1 2

2 1 ,

21 .

pp

a

p rm GMtF F r r r

rm GMtF r r r

∆ ∆ = = − +

∆ = − +

3 2 .2p aPt a

GMπ

−∆ = =


Asteroids: the reverse case

Close encounters between asteroids, perhaps driven by planetary perturbations on asteroids that lie in mean-motion resonances, can result in sudden losses of momentum and energy for one of the bodies participating in the encounter. That latter body can thus wind up in a highly eccentric

orbit that crosses the orbits of other asteroids and planets. In such orbits an asteroid is even more vulnerable to close

encounters, which can result in destruction or scattering into other, lower-eccentricity, orbits inward from the original orbit.

This process of naturally-occurring transfer orbits continually rearranges the small bodies in the Solar system.

Asteroids: the reverse case (continued)

Example. An asteroid initially in a circular orbit at 2.5 AU suffers a collision that knocks it into an orbit tangent to the Earth’s. (a) Describe the change in its velocity, caused by the collision.


( ) ( )

ε

ε ε

−

−

= = = =

= = − = = +

+ ⇒ = = = − =

11 1 1

1

2 2 2 2

12 2

2

Before:

2.5AU 0 18.8 km sec

After:1AU 1 2.5AU 1

2 11.75AU, 14.2 km sec2

p a

p a

a

GMa v

a

r a r a

r ra v GM

r a

Asteroids: the reverse case (continued)

So the collision slows the asteroid down by 4.6 km sec-1. (b) What sort of collision would cause this change? Could a small orbital change by a small body in a nearby orbit do this?Suppose it’s a completely inelastic collision with a much smaller body:

So: it can’t, really. A small body would have to be revolving in the opposite direction of everything else to cause such an orbital change in our original asteroid. This is why orbital changes by asteroid encounters are usually small and need to add up over a long time to amount to much.


( )

( )

+ = + ≈

≈ − <

1 2 2

2 1 0

c c c

cc

mv m v m m v mvmv v vm


The two-body gravitational potential: introduction to the restricted three-body problem

Consider a three-body orbiting system in which one body has negligibly small mass, compared to the other two. It is reasonable to construct the orbital energy of the

system out of the energies of the two massive components, and consider the third body simply to follow along. • Define the gravitational potential, Φ, as the potential

energy per unit mass for the smaller body. It is also reasonable to do this in a coordinate system that

rotates with the revolution of the two massive bodies.• The potential energy will contain terms corresponding

to “fictitious,” inertial forces such as centrifugal force and Coriolis force.


The two-body gravitational potential (continued)

1 2 1 1 2 22 2 21 1 12 2 22 2 2

1 2centr

1 2

centr

In corotating frame of reference, neglecting Coriolis force:

2 cos

2 cosGravitational potential:

,

where is the potential due t

r r a M r M r

s r r r r

s r r r r

M MGs s

θ

θ

+ = =

= + +

= + −

Φ = − + +Φ

Φ

o centrifugal forces felt at . r

y

x

s2s1

r1 r2 M2M1

r

θCM

a



Gravitational potential energy is the work one must do against the force of gravity if one moves a body in a gravitational field.

Analogously, the centrifugal potential is the work per unit mass done against the centrifugal force when one moves a body around with respect to a rotating coordinate system:

2centr centr

2 212

m W d m rdr

m r

ω

ω

Φ = = − ⋅ = −

= −

∫ ∫f f

i i

r r

r rF r



By Kepler’s third law,

so the gravitational potential becomes

( )

2 21 2

1 2

1 2 21 23

1 2

12

.2

M MG rs s

G M MM MG rs s a

ω

Φ = − + −

+ = − + −

( )2 1 223

2 ,G M M

P aπω

+ = =



Plot of

as a function of and Contours are loci of constant gravitational potential; the purple colors show relative maxima of potential.

L1L2L3

L4

L5

M1

CM M2

0 2-2-2

0

2

x (AU)

y(A

U)

M MM Ma

1

2

= 5= 1

= 1 AU

( )

1 2

1 2

1 2 232

M MGs s

G M Mr

a

Φ = − +

+

−

sin .y r θ=cosx r θ=


Lagrange points

Extrema of the gravitational potential, of which there are five besides the massive bodies, are called Lagrange points. Gravitational force vanishes

there; an object placed at one will stay fixed there, and orbit the CM along with the stars…

unless they suffer perturba-tions: they are maxima of Φ.

Thus spacecraft placed thereneed thrust to stay there(though not much); natural satellites will not collect there.

L1L2L3

L4

L5

M1

CM M2

0 2-2-2

0

2

x (AU)

y(A

U)

M MM Ma

1

2

= 5= 1

= 1 AU


Lagrange points (continued)

Exceptions: if Coriolis forces are included, and one of the massive bodies outweighs the other by a factor of 25 or more, it turns out that the small body can stably orbit the fourth or fifth Lagrange points.• It must orbit L4 or L5; it can’t sit stably exactly at L4 or

L5. But stable, naturally-occurring collections of particles can orbit there.

• Such orbits are called halo orbits.• In analogy with two groups of asteroids trapped in

Jupiter’s L4 and L5 points, collections of particles in halo orbits about the fourth and fifth Lagrange points are called Trojans.

The fourth and fifth Lagrange points

These points turn out to lie at the apices of the two equilateral triangles which can be drawn in the orbital plane using the line between the two massive objects as one side. Since the Sun is more than

25 times as massive as any of the planets, stable halo orbits about L4 and L5 could accompany any planet orbit. (At least the nearly circular ones.)


L1L2L3

L4

L5

M1

CM M2

0 2-2-2

0

2

x (AU)

y(A

U)

M MM Ma

1

2

= 5= 1

= 1 AU


Halo orbits about the fourth and fifth Lagrange points

Simulation of asteroids locked around L4 and L5 in a 1:1 mean-motion resonance with Earth, by Paul Wiegert (U. W. Ontario) and colleagues.

http://www.astro.uwo.ca/~wiegert/etrojans/etrojans.html�http://www.astro.uwo.ca/~wiegert/�

Today in Astronomy 111: multiple-body systemsHohmann transfer orbitsHohmann transfer orbits (continued)Hohmann transfer orbits (continued)TerminologyHohmann transfer orbits (continued)Hohmann transfer orbits (continued)Hohmann transfer orbits (continued)Asteroids: the reverse caseAsteroids: the reverse case (continued)Asteroids: the reverse case (continued)The two-body gravitational potential: introduction to the restricted three-body problemThe two-body gravitational potential (continued)The two-body gravitational potential (continued)The two-body gravitational potential (continued)The two-body gravitational potential (continued)Lagrange pointsLagrange points (continued)The fourth and fifth Lagrange pointsHalo orbits about the fourth and �fifth Lagrange points

Documents

Today in Astronomy 111: introduction to three-body systemsdmw/ast111/Lectures/Lect_11b.pdfToday in Astronomy 111: multiple- body systems Two simple topics in systems of three bodies: