Planet Formation - PowerPoint Presentation

Slide1

Planet Formation

Topic:

Orbital dynamics and

the restricted 3-body problem

Lecture by: C.P. DullemondSlide2

Objects of the Solar System

Kuiper Belt

(30-50 AU)

10 AU

1 AU

Earth

Mars

1 AU

1.38-

1.67 AU

e

= 0.1

0.72 AU

Venus

0.30-

0.47 AU

e

= 0.2

Asteroid Belt (2.0-3.4 AU)

Mercury

4.95-5.46 AU

e

= 0.05

Jupiter

Saturn

9.05-10.1 AU

e

= 0.05

18-20 AU

e

= 0.05

Uranus

Neptune

3

0 AU

Logarithmic distance scale

29.6-

48.9 AU

e

= 0.25

Pluto

Ceres

2.55-

2.99 AU

e

= 0.08

Planets

Dwarf-

Planets

Asteroids

& TNOs

+

Haumea

,

Makemake,Eris

Rocky Planets

Gas Giants

Ice GiantsSlide3

Kepler orbits, eccentricity

Kreisbahn

(

Mplanet << M*

):

(Kepler Frequency)

(Kepler

Velocity)

(„True

anomaly

“)

Sonne

a

Planet

x

ySlide4

Kepler orbits, eccentricityElliptic orbit

(

Mplanet << M

*):

(Kepler Freq.)

(„

Mean

anomaly

“)

(„

Eccentric

anomaly

“ ,

solve

numerically

)

(„True

anomaly

“)

a („Semi-major

axis

“)

e

(„

eccentricity

“)

Sonne

a

Planet

e

a

r

Periapsis

(

Perihelion

)

Apoapsis

(

Aphelion

)

Fokus

x

ySlide5

Kepler orbits, eccentricityElliptic orbit

(

Mplanet << M

*):

(Kepler Freq.)

Sonne

a

Planet

e

a

r

Periapsis

(

Perihelion

)

Apoapsis

(

Aphelion

)

Fokus

x

y

Total energy:

Note: Total energy depends only on

a

!

Angular momentum:

Note: eccentricity thus leads to an

angular momentum deficit

.Slide6

Orbital elements

From: Wikipedia: http://en.wikipedia.org/wiki/File:Orbit1.svg

Compared to the

equatorial plane

of the solar system

(or exoplanetarysystem), and compared to a reference direction,you can uniquelyorient the ellipse.

Semi-major axis a,eccentricity e andtrue anomaly

ν, together with orientation (i, ω, Ω)are: orbital elements.Slide7

Guiding-center and epicyclic motionIf an orbit is almost

circular, then we can describe this elliptic orbit

as a circular orbit with epicyclic motion superposed on it.

e=0.3

e=0.3Slide8

Guiding-center and epicyclic motionWhen using another orbital frequency as the reference frame than

the orbital frequency of the particle, this epicyclic motion looks like:

e=0.3

e=0.3Slide9

Guiding-center and epicyclic motion

guiding

center

epicycle

For small eccentricity

the epicycle becomes

an ellipse.

As we will see (and use)

later: these small

deviations from Kepler

can be described as

velocity disturbances

Δv.Slide10

Even kids are familiar with this... ;-)Slide11

What is the „restricted 3-body problem“?Three bodies, one of which (M3) is a „test particle“:

M

3

<<< M2 < M1Bodies 1 and 2 only feel each other‘s gravity and thus perfectly follow a Kepler orbit.

Assume that body 1 and 2 are in perfect circular orbitsPut the center of coordinate system at center of massBody 3 feels bodies 1 and 2.Resulting motions:Some orbits are stable

Some orbits are unstable (body 3 gets ejected)Some orbits are chaotic: Chaos theory!Chaotic orbits are unpredictable on the long run.Slide12

Equations of motion for test particle

Remember from chapter „Turbulence“ Section „Magnetorotational

instability“ the equation of motion of a test particle in a rotating frame:

(from chapter

„Turbulence“)

Now we do the same (though now we put x=0 at the center of

mass of the entire system), we drop the f

x and fy forces but now

we include the forces of both the star and the planet.

Coriolis forces

Gravity

and

centrifugal forces

With the „effective potential“ given by:

Exercise: re-derive these equations.Slide13

Effective potential, Lagrange points

r

1

r

2

Effective potential in the

co-rotating frame:

centrifugal kinetic energy

Example: M

2

/M

1

=0.1

L

1

L

2

L

3

L

4

L

5

Gravitational

potentialSlide14

Effective potential, Lagrange points

r

1

r

2

Effective potential in the

co-rotating frame:

Example: M

2

/M

1

=0.01

L

1

L

2

L

3

L

4

L

5

centrifugal kinetic energy

Gravitational

potentialSlide15

Jacobi‘s IntegralThe full 3-D set of equations is:

now multiply by:

and add them all up:Slide16

Jacobi‘s Integral

This can be integrated once, to obtain:

Traditionally the constant

C

is written as

-½C

J

:

C

J

is called Jacobi‘s constant or Jacobi‘s Integral of motion.

For the restricted 3-body problem it is the only integral of motion,i.e. there exist no closed-form solutions. Slide17

Zero-velocity curves / surfacesJacobi‘s constant is some kind of energy, sometimes called„Integral of relative energy“. It is the rotational equivalent of minus

twice the total (potential + kinetic) energy of a test particle in a non-

rotating system:

>0

<0

Since the kinetic term <0, we know that a given particle

on a given orbit (with a given constant C

J

), can only reach

points (x,y,z) where the effective potential obeys:

Remember:Slide18

Zero-velocity curves / surfaces

(allowed region in x,y,z)

The boundaries of this region are called the „zero velocity curves“ (in

2-D) or „zero velocity surfaces“ (in 3-D). They are the potential lines.

For C

J

<0 no such restrictions exist (all points are allowed). More

precisely: for C

J

<min(-2Φ

eff

) no such restriction exist. But for

C

J

>min(-2Φeff) there exists regions in (x,y,z) which are inaccessiblefor the particle. If CJ

is sufficiently large, these inaccessible regions can even completely surround the star or planet system or both:

= Not allowed regionSlide19

Not exactly a sphere, but approximately.

It is the largest zero-velocity surface

surrounding

only

the planet.

It is the

sphere of influence

of the planet.

Hill sphere (=Roche lobe)Slide20

Meaning of Hill sphereThe Hill sphere plays a key role in planet formation:To add mass to a planet, we must put the mass into the Hill sphere of the growing planet, because only then it that mass gravitationally bound to it.

Any circumplanetary disk or moons must be inside the planet‘s Hill radius

We will see later that the ratio of the Hill radius to the protoplanetary disk‘s thickness plays a key role in planet migration.

Any object that is larger than its own Hill sphere will be sheared apart by the tidal forces of the star. This will lead us to the definition of the „Roche density“ as the minimal density an object needs to remain gravitationally coherent and survive tidal forces.

See next chapter.Slide21

Potential field lines as approximate orbitsIf then one can approximately write:

This means that

approximately

the test particle moves along

the potential field lines. To be more precise: the guiding center

will do this; the test particle will epicycle around this guiding

center. You can find orbits without epicycles, in which case the

test particle indeed moves approximately along the field lines.

It turns out to be also a fairly good approximation if the condition

does not hold.This leads to a special set of orbits.Slide22

Kepler- and horseshoe orbits

Kepler orbit

around star only

Kepler orbit

around planet only

Kepler orbit

around both

Horseshoe orbitSlide23

L4 and L5 are stable Lagrange points, while L1, L2 and L3 are not.

Trojans of Jupiter