More Orbital Dynamics

❑Elliptical orbits

❑Mean motion resonances

❑Secular resonances

❑Hohmann transfer orbit

Ast111, Lecture 2b

Elliptic paths222

2

2 2

2

2

2 2 2

4 2

Energy and angular momentum are conserved

2 2 2

2 2We want a formula for ( )

2 2For

r

r

r

vvE GM GMm r rL dl rv rm dt

vE l GMm r r

rdr dr d dr lvdt d dt d r

E dr l l GMm d r r r

θ

θ

θ

θ

θ

θ θ

θ

= − = + −

= = =

= + −

= = =

# $= + −% &' (

v

0, the solution ( ) to this is an ellipse.E r θ<

It is useful to divide the velocities into radial and azimuthal components. This makes it easy to calculate quantities at perihelion and aphelion because there we know that vr=0.

It is often useful to show these equations in terms of Energy and Angular momentum per unit mass.

If you solve this differential equation you will find conic section solutions``

Kepler’s First Law2

1/ 2

(1 ) where is the semi-major axis

1 cos, the true anomaly is the angle between the planet's

perhelion and its position (like with respect to the Sun

he eccentricity, (1 / )

a er ae f

f

e b a

E

θ

−=

+

) .

Τ = −

= −

2 3

2

is the energy of the orbit.2

The orbital period is set by .

(in years)= (in AU).

(1 ) is the angular momentum.

GMa

aP a

L GMa e= −

Energy and Angular momentum of an elliptical orbit

2

/2

/ (1 )For a given energy, the orbit with the maximum angularmomentum is a circular one.If there are additional forces on the object, the balance between torque (affecting angular momet

GME ma

L m GMa e

= −

= −

um) and energyloss, will determine how the orbital elements change.

Planetary Perturbations

• Purely Keplerian motion assumes a perfect inverse square force law.

• When there is any deviation from the perfect inverse square law, orbits are no longer perfect ellipses.

• Another way to say this: the orbits don’t exactly close.

Precession of orbitsBecause of forces from other planets, the angle of perihelion of the planets slowly change.

We say the orbit precesses.

The angle of perihelion of Mercury changes because of General relativistic effects which also make the gravitational potential deviate from a perfect inverse square law.

Perturbation of orbits by other planets

❑ Because of perturbation and slight departure from closed elliptical orbits, the orientations of planetary orbits slowly change. We call this effect perihelion precession.

❑ The best example is Mercury, with a perihelion precession of 5600.73 arcsec/century. 5025 arcsec/century of this is an artifact of our coordinate system (the precession of equinoxes, due to Earth’s rotational-axis precession, but the remaining 576 arcsec/century is due to perturbations: 280 arcsec/century from Venus 151 arcsec/century from Jupiter 102 arcsec/century from other planets (mostly Earth) 43 arcsec/century from general relativity (warping of space by the Sun)

Stellar orbits in Galaxies • Because matter is spread

out over a large region in galaxies, the force law is very far from inverse square.

• Orbits typically form rosette patterns.

Radial variations in orbit

Nearly circular orbits ! Epicyclic approximation

E � v2r2

=L2

2r2� GM

r

1

2r2� 1

r

Fix L but adjust E

distance is vr2/2

pericenter apocenter

The 3-body problem

• The two body problem is “completely integrable.” That means that there is one conserved quantity per degree of freedom.

• Orbits can be predicted exactly for the 2-body system.

• For the 3-body problem (3 interacting bodies), there is no general analytical solution.

• Even the “restricted 3-body problem,” (two massive particles and one massless), is complicated enough to exhibit resonant and chaotic phenomena.

ResonancesPeriodic perturbations on an object, such as the nearby passage of a planet, will add up if they are in phase with the orbital motion of the object.

If there are out of phase, then the perturbations will cancel on average and there will be no net change in the orbit of the object.

Small Perturbations can be important

• The solar system is about 4 billion years old. • This means that there have been about 109

orbits. A billion small perturbations, if they add up, can really push around a particle or asteroid.

Mean Motion Resonances• Consider the force caused by a planet on a nearby particle. • When do perturbations from the planet add up? • One condition is when and integer times the orbital period of

the planet is equal to an integer times the orbital period of the particle.

n =

rGM

a3

n =2⇡

PMean motion n Orbital period P semi-major axis a n is also the angular rotation rate if the orbit is circular

Mean Motion Resonances

Condition for resonance where , are integers,

is the mean motion of the planet and

is the mean motion of the object.We can also write this condition as

in terms of the orbit period

p

p

p

in jn i jnn

jP iP

≈

≈ s.

We identify the resonance as the : resonance.i j

Where are Mean Motion Resonances located?

3/ 2

2/3

Remember that the orbital period

2

So a condition for a mean motion resonance is that

where , refer to the semi-major axes

of the object and plane

pp

aPGM

a i a aa j

π=

" # " #≈% & % &% & ' (' (

t, , are integers.i j

Resonant angle

In the frame rotating with the planet

φ λ λ π

− + =

= − + =

( 1) 0

mean motions are integer multiples( 1) constant like 0 or

remains fixed

p

p

jn j n

j j

Resonant angle

Librating resonant angle "!in resonance

Oscillating resonant angle "!outside resonance

The main asteroid belt

Snapshot of distribution of main-belt asteroids, as would be viewed from the ecliptic pole (Bidstrup et al. 2005). The Sun’s position is labelled with a blue star, and the five inner planets by red blocks.

Distance from Sun (AU)

Dis

tanc

e fr

om S

un (A

U)

Distance from Sun (AU)

Num

ber o

f ast

eroi

ds

Mean Motion Resonances are responsible for the Kirkwood Gaps

The main asteroid belt (continued)

Snapshot of distribution of main-belt asteroids above and below the ecliptic (left), and perspective sketch of volume occupied by the main belt (right) (Bidstrup et al. 2005).

Distance from Sun (AU)

Dis

tanc

e ab

ove

eclip

tic (A

U)

Resonances as a source of instability

• In the inner asteroid belt, particles knocked into strong mean motion resonances with Jupiter, slowly increase in eccentricity until they cross the orbit of a planet, and can be scattered out of the system.

• Scattered objects can become comets, go into the Oort cloud, or even become sungrazers or impactors.

• We don’t really know how many things were tossed out of our Solar system, because all we see is an old remnant.

Resonances can also be a place of stability

Hildas, Thule and Trojan groups of asteroids are locked in resonances with Jupiter

Resonances resulting in stability

• The Trojans, Hildas and Thule asteroids are in orbits such that they always avoid Jupiter. In this way they are never knocked out of the system.

Pluto is in a 3:2 mean motion resonance with Neptune

Even though Pluto’s orbit goes inside Neptune’s orbit, it never is scattered by Neptune because every time Pluto reaches perihelion, Neptune is in a different position.

Frame rotating with Neptune

Palas Jupiter

Haumea Neptune

Plutinos and twotinos in the Kuiper belt

Red: resonant Blue: classical Grey: scattered

incl

inat

ion

Multiple bodies in resonance

Laplace resonance

Resonances in Planetary Rings

Spiral density waves in Saturn's A Ring excited by resonances with inner moons. Such waves propagate away from the planet (towards upper left). The large set of waves just below center is due to the 6:5 resonance with Janus.

The Cassini-Huygens Mission • Much of the structure in

Saturn’s rings is due to resonances with satellites.

• Not all the structure has been explained (though lots of interest processes including coupling with internal oscillations and perturbations by passing comets).

Secular Timescales• When something takes place on a timescale much longer

than the rotation period, we say it takes place on a “secular” timescale.

• Precession of orbits typically takes more than a thousand orbits.

• Sometimes it is mathematically convenient to average over the motion of the orbit.

• A secular approximation involves averaging over the entire orbit. It is equivalent to spreading the mass of a planet out around its orbit.

• If the planet is in an elliptical orbit, the force from it can exert a torque on other objects.

Secular Resonances

• Because planets precess, there can be resonances between the precession period of an object and a planet. These are called secular resonances and can involve eccentricity or inclination changes.

• In the asteroid belt there is a strong resonance with the precession rate of Saturn.

• This resonance also causes objects to graze or impact the Sun and has sculpted the asteroid belt (edge at 2.1 AU).

Pendulum

contours of energy

l = length g = gravitational acceleration

m✓̈ = �m

rg

lsin ✓

E

m=

˙✓2

2

� g

lcos ✓

Pendulum types of motion

• Oscillation: angle continues to increase • Libration: angle remains near a constant

Resonance Overlap and Chaotic Motion

• When more than one resonance affects an object, the resulting motion can be “chaotic.”

• A simple analogy is the forced or double pendulum which is a well known chaotic system.

Forced pendulum that can display chaotic motion

E

m=

1

2

˙✓2 � g

l(cos ✓ + µ cos(✓ � ⌫t))

Chaotic trajectories for the restricted 3 body problem

These are solutions to the three-body gravitational problem which differ only in that their initial horizontal speeds were different by one percent. Initially, the two "planets" track each other almost perfectly, so you can only see one of them. Later the trajectories diverge.

Resonance Overlap near a planet

• Characteristics of chaotic orbits include extreme sensitivity to initial conditions, exponential divergence of orbits away from each other.

• At radii closer and closer to a planet, mean motion resonances get closer and closer. Eventually there is a chaotic region where all these resonances overlap.

• In the chaotic band, motion cannot be separated between the effects of individual resonances.

Hohmann transfer orbit

• Moving from one circular orbit to another one with two instantaneous changes in velocity.

• From a geo-synchronous orbit to another orbit

• From Earth to Mars orbit. To reach Mars itself – “launch window” in time

Calculating Δv

Vis-viva equation (Latin for "live force")

E = �GM

2a=

v2

2� GM

r

v2 = GM

✓2

r� 1

a

◆

How do you find Δv? and Δv’?

Δv in tangential direction A change in angular momentum In both transfer spots angular momentum is increased

Calculating ΔvE = �GM

2a=

v2

2� GM

r

v2 = GM

✓2

r� 1

a

◆

ratio of minimum to maximum radius for an orbit (1-e)/(1+e) Transfer orbit has to have apocenter at R’ where R’=a(1+e) pericenter at R where R =a(1-e) Can solve for eccentricity and semi-major axis of transfer orbit Then from this solve for 𝞓v and 𝞓v’

Calculating Δv

• Summing in Δv for total thrust needed • Larger Δv required if using more than one

transfer orbit.

�v =

rµ

R

r2R0

R+R0 � 1

!

�v0 =

rµ

R0

1�

r2R

R+R0

! µ = GM

Review222

2

2

Energy and Angular momentum conservation

2 2 2

(1 )

(1 )Shape of orbit:

1 cosMean motion resonanc

r vvE GM GM E GMm r r m a

LL mrv GMa em

a ere f

θ

θ

= − = + − = −

= = −

−=

+

v

1 2

1 2

es at Where , are integers, and , are mean motions

in jni j n n

≈

Vocabulary and Concepts: Orbital precession, Secular timescale, Resonances, Resonance Overlap, Kirkwood Gaps, Resonance Capture, Chaotic dynamics, mean motion

3

2 GMnP aπ

= =

E

m= �GM

2aL

m=

pGMa(1� e2)

E

m=

v2

2� GM

r=

v2r2

+v2✓2

� GM

r

Keplerian orbits, energy + angular momentum conservation

L/m = rv✓ = r2✓̇

r =

a(1� e2)

1 + e cos forbit shape mean motion