UofT. Lecture L5 - Scenarios of Planet formation 0. Student presentation by Libby 1. Top-down (GGP hypothesis) and its difficulties 2. Standard (core-accretion) scenario ( a ) from dust to planetesimals - two ways ( b ) from planetesimals to planetary cores: how many planetesimals were there? ( c ) Safronov number, runaway growth, oligarchic growth of protoplanets

Or some other way??

Note: the standard scenario on the left also looks like the r.h.s. pictures…. With one major difference: time of formation of giant protoplanets: 3-10 Myr (left panel) 0.1 Myr (right panel)

There are two main possible modes of formation of giant gaseous planets and exoplanets: # bottom-up, or accumulation scenario for rocky cores (a.k.a. standard theory) predicts formation time ~(3-10) Myr (V.Safronov, G.Kuiper, top-down, by accretion disk breakup as a result of gravitational instability of the disk. A.k.a. GGP = Giant Gaseous Protoplanets formation time < 0.1 Myr (I.Kant, G.Kuiper, A.Cameron) To understand the perceived need we have to consider disk evolution and observed time scales: cores may not be ready! To understand the physics we need to study the stability of disks against self-gravity waves.

Gravitational Instability and the Giant Gaseous Protoplanet hypothesis

Stability vs. fragmentation of disks

Self-gravity as a destabilizing force for the epicyclic oscillations (radial excursions) of gas parcels on slightly elliptic orbits To study waves in disks, we substitute into the equations of hydrodynamics the wave in a WKBJ (a.k.a. WKB) approximation, also used in quantum mechanics: it assumes that waves are sinusoidal, tightly wrapped, or that kr >>1. All quantities describing the flow of gas in a disk, such as the density and velocity components, are Fourier-analyzed as Gregor Wentzel ( ) German/American physicist Hendrick A. Kramers (1894–1952) Dutch physicist Léon N. Brillouin ( ) French physicist Harold Jeffreys (1891 –1989) English mathematician, geophysicist, and astronomer, established a general method of approximation of ODEs in Some history WKB applied to Schrödinger equation (1925),

An example of a crest of a spiral wave ~X 1 exp[ … ] for k = const >0, m = 2, = const. This spiral pattern has a constant shape and rotates with an angular pattern speed equal to The argument of the exponential function, is constant on a spiral wavecrest

Dispersion Relation for non-axisymmetric waves in disks* tight-winding (WKB) local approximation Doppler-shifted frequency epicyclic frequency self- gas gravity pressure In Keplerian disks, i.e.disks around point-mass objects * - for derivation see Binney and Tremaine’s book (1990) “Galactic Dynamics”

As elsewhere in physics, the dispersion relation is the dependence between the time- and spatial frequencies, Though it looks much more frightening than the one describing the simple harmonic (sinusoidal) plane wave of sound in the air: you can easily convince yourself that in the limit of vanishing constant G (no self-gravity in low-mass disks!) and vanishing epicyclic frequency (no rotation!), the full dispersion relation assumes the above form. Therefore, the waves in a non-rotating medium w/o gravity are simply pure pressure (sound) waves. The complications due to the differential rotation lead to a spiral shape of the sound- or the fully self-gravitating density wave.

Dispersion Relation in disks with axisymmetric (m=0) waves (1960,1964)

Gravitational stability requirements Local stability of disk, spiral waves may grow Local linear instability of waves, clumps form, but their further evolution depends on equation of state of the gas.

Question: Do we have to worry about self-gravity and instability? Ans: Yes Ans: No z/r~0.1 q d >0.1

Clumps forming in a gravitationally unstable disk (Q < 1) Recently, Alan Boss revived the half-abandoned idea of disk fragmentation GGPs?

From: Laughlin & Bodenheimer (2001) Disk in this SPH simulation initially had Q ~ 1.5 > 1 The m-armed global spiral modes of the form grow and compete with each other. But the waves in a stable Q~2 disk stop growing and do not form small objects (GGPs). It turns out that even at Q~1.5 there are unstable global modes.

Two examples of formally unstable disks not willing to form objects immediatelyDurisen et al. (2003) Break-up of the disk depends on the equation of state of the gas, and the treatment of boundary conditions.

Simulations of self-gravitating objects forming in the disk (with grid-based hydrodynamics) shows that rapid thermal cooling is crucial Armitage and Rice (2003) Disk not allowed to cool rapidly (cooling timescale > 1 P) Disk allowed to cool rapidly (on dynamical timescale, <0.5 P)

SPH = Smoothed Particle Hydrodynamics with 1 million particles Mayer, Quinn, Wadsley, Stadel (2003) Isothermal (infinitely rapid cooling)

GGP (Giant Gaseous Protoplanet) hypothesis = disk fragmentation scenario (A. Cameron in the 1970s) Main Advantages: forms giant planets quickly, avoids possible timescale paradox; planets tend to form at large distances amenable to imaging. MAIN DIFFICULTIES: 1. Non-axisymmetric and/or non-local spiral modes start developing not only at Q<1 but already when Q decreases to Q~1.5…2 They redistribute mass and heat the disk => increase Q (stabilize disk). 2. Empirically, this self-regulation of the effects of gravity on disk is seen in disk galaxies, all of which have Q~2 and yet do not split into many baby gallaxies. 3. The only way to force the disk fragmentation is to lower Q~c/Sigma by a factor of 2 in just one orbital period. This seems impossible. 4. Any clumps in disk (a la Boss’ clumps) may in fact shear and disappear rather than form bound objects. Durisen et al. Have found that the equation of state and the correct treatment of boundary conditions are crucial, but could not confirm the fragmentation except in the isothermal E.O.S. case. 5. GGP is difficult to apply to Uranus and Neptune; final masses: Brown Dwarfs not GGPs 6. Does not easily explain core masses of planets and exoplanets, nor the chemical correlations

OBSERVATIONS OF CORES IN EXOPLANETS

Comparison of gas and rock masses (in M E ) in giant planets and exoplanets (1980s) Planet Core mass Atmosph. Total mass Radius _________(rocks, M E )___(gas,_M E ) ____(M E ) _______( R J ) _ Jupiter 0-10 ~ Saturn ~ Uranus Neptune core envelope (atmosphere)

Comparison of gas and rock masses (in M E ) in giant planets and exoplanets (Oct. 2005) Planet Core mass Atmosph. Total mass Radius _________(rocks, M E )___(gas,_M E ) ____(M E ) _______( R J ) _ Jupiter 0-10 ~ Saturn ~ Uranus Neptune HD b ~0 ~ ± 0.05 (disc. 1999) HD b ~ 70 ~ ± 0.03 (disc. 7/2005) HD b ~10-20(?) ~ ± 0.03 (disc. 10/2005) core envelope (atmosphere) ? ?

Video of density waves in a massive protoplanetary disk The shocks at the surface are suggested as a way to heat solids and form chondrules, small round grains inside meteorites. Durisen and Boss (2005) …but that is another issue.

Standard Accumulation Scenario

Two-stage accumulation of planets in disks

M core =10 M E (?) => contraction of the atmosphere and inflow of gas from the disk Planetesimal = solid body >1 km (issues not fully addressed in the standard theory, so far)

Two scenarios proposed for planetesimal formation Particles settle in a very thin sub-disk, in which Q<1, then gravitational instability in dust layer forms planetesimals Particles in a turbulent gas not able to achieve Q<1, stick together via non-gravity forces. gas Solid particles (dust, meteoroids) gas Q>1 Q<1

How many planetesimals formed in the solar nebula?

Disk mass ~ 0.02 … 0.1 Msun > 2e33g*0.02=4e31 g ~ 1e32 g Dust mass ~ (0.5% of that) ~ 1e30 g ~ 100 Earth masses (Z=0.02 but some heavy elements don’t condense) Planetesimal mass, s =1 km = 1e5 cm, ~(4*pi/3) s^3*(1 g/cm^3) => 1e16 g Assuming 100% efficiency of planet formation N = 1e30g /1e16g = 1e14, s=1 km N = 1e30g /1e19g = 1e12, s=10 km (at least that many initially) N = 1e30g /1e22g = 1e8, s=100 km N = 1e30g /1e25g = 1e5, s=1000 km N = 1e30g /1e28g = 100, s=10000 km (rock/ice cores & planets) N = 1e30g /1e29g = 10, (~10 M E rock/ice cores of giant planets)

Gravitational focusing factor

Oligarchs rule their vicinity but do not interfere with each other

Non-perturbative methods (energy constraints, integrals of motion, Roche Lobe, stability of orbits) Karl Gustav Jacob Jacobi ( ) Joseph-Louis Lagrange ( ) This following 10+ slides are a digression on celestial mechanics:

A standard trick to obtain energy integral Solar sail problem again

Energy criterion guarantees that a particle cannot cross the Zero Velocity Curve (or surface), and therefore is stable in the Jacobi sense (energetically). However, remember that this is particular definition of stability which allows the particle to physically collide with the massive body or bodies -- only the escape from the allowed region is forbidden! In our case, substituting v=0 into Jacobi constant, we obtain:

f=0 f=0.051 < (1/16) f=0.063 > (1/16) f=0.125 Allowed regions of motion in solar wind (hatched) lie within the Zero Velocity Curve particle cannot escape from the planet located at (0,0) particle can (but doesn’t always do!) escape from the planet (cf. numerical cases B and C, where f=0.134, and 0.2, much above the limit of f=1/16 ).

Circular Restricted 3-Body Problem (R3B) L1L1 L4L4 L5L5 L3L3 L2L2 “Restricted” because the gravity of particle moving around the two massive bodies is neglected (so it’s a 2-Body problem plus 1 massless particle, not shown in the figure.) Furthermore, a circular motion of two massive bodies is assumed. General 3-body problem has no known closed-form (analytical) solution. Joseph-Louis Lagrange ( ) [born: Giuseppe Lodovico Lagrangia]

NOTES: The derivation of energy (Jacobi) integral in R3B does not differ significantly from the analogous derivation of energy conservation law in the inertial frame, e.g., we also form the dot product of the equations of motion with velocity and convert the l.h.s. to full time derivative of specific kinetic energy. On the r.h.s., however, we now have two additional accelerations (Coriolis and centrifugal terms) due to frame rotation (non-inertial, accelerated frame). However, the dot product of velocity and the Coriolis term, itself a vector perpendicular to velocity, vanishes. The centrifugal term can be written as a gradient of a ‘centrifugal potential’ -(1/2)n^2 r^2, which added to the usual sum of -1/r gravitational potentials of two bodies, forms an effective potential Phi_eff. Notice that, for historical reasons, the effective R3B potential is defined as positive, that is, Phi_eff is the sum of two +1/r terms and +(n^2/2)r^2

R3B =

Effective potential in R3B mass ratio = 0.2 The effective potential of R3B is defined as negative of the usual Jacobi energy integral. The gravitational potential wells around the two bodies thus appear as chimneys.

Lagrange points L 1 …L 5 are equilibrium points in the circular R3B problem, which is formulated in the frame corotating with the binary system. Acceleration and velocity both equal 0 there. They are found at zero-gradient points of the effective potential of R3B. Two of them are triangular points (extrema of potential). Three co-linear Lagrange points are saddle points of potential.

r L = Roche lobe radius + Lagrange points Jacobi integral and the topology of Zero Velocity Curves in R3B

Sequence of allowed regions of motion (hatched) for particles starting with different C values (essentially, Jacobi constant ~ energy in corotating frame) Highest C Medium C High C (e.g., particle starts close to one of the massive bodies) Low C (for instance, due to high init. velocity) Notice a curious fact: regions near L 4 & L 5 are forbidden. These are potential maxima (taking a physical, negative gravity potential sign)

Édouard Roche (1820–1883), Roche lobes terminology: Roche lobe ~ Hill sphere ~ sphere of influence (not really a sphere) = 0.1 C = R3B Jacobi constant with v=0

Is the motion around Lagrange points stable? Stability of motion near L-points can be studied in the 1st order perturbation theory (with unperturbed motion being state of rest at equilibrium point).

Stability of Lagrange points Although the L 1, L 2, and L 3 points are nominally unstable, it turns out that it is possible to find stable and nearly-stable periodic orbits around these points in the R3B problem. They are used in the Sun-Earth and Earth-Moon systems for space missions parked in the vicinity of these L-points. By contrast, despite being the maxima of effective potential, L 4 and L 5 are stable equilibria, provided M 1 /M 2 is > (as in Sun-Earth, Sun-Jupiter, and Earth-Moon cases). When a body at these points is perturbed, it moves away from the point, but the Coriolis force then bends the trajectory into a stable orbit around the point.

From: Solar System Dynamics, C.D. Murray and S.F.Dermott, CUP Observational proof of the stability of triangular equilibrium points Greeks, L 4 Trojans, L 5

= 0.1 = 0.01 Roche lobe radius depends weakly on R3B mass parameter

Computation of Roche lobe radius from R3B equations of motion (, a = semi-major axis of the binary) L

= 0.1 = 0.01 Roche lobe radius depends weakly on R3B mass parameter m 2 /M = 0.01 (Earth ~Moon) r_L = 0.15 a m 2 /M = (Sun- 3xJupiter) r_L = 0.10 a m 2 /M = (Sun-Jupiter) r_L = 0.07 a m 2 /M = (Sun-Earth) r_L = 0.01 a

George W. Hill ( ) - studied the small mass ratio limit of in the R3B, now called the Hill problem. He ‘straightened’ the azimuthal coordinate by replacing it with a local Cartesian coordinate y, and replaced r with x. L 1 and L 2 points became equidistant from the planet. Other L points actually disappeared, but that’s natural, since they are not local (Hill’s equations are simpler than R3B ones, but are good approximations to R3B only locally!) Roche lobe ~ Hill sphere ~ sphere of influence (not really a sphere, though) Hill problem

Hill applied his equations to the Sun-Earth-Moon problem, showing that the Moon’s Jacobi constant C= is larger than C L = (value of effective potential at the L-point), which means that its Zero Velocity Surface lies inside its Hill sphere and no escape from the Earth is possible: the Moon is Hill-stable. However, this is not a strict proof of Moon’s eternal stability because: (1) circular orbit of the Earth was assumed (crucial for constancy of Jacobi’s C) (2) Moon was approximated as a massless body, like in R3B. (3) Energy constraints can never exclude the possibility of Moon-Earth collision Hill problem

How wide a region is destabilized by a planet?

Hill stability of circumstellar motion near the planet The gravitational influence of a small body (a planet around a star, for instance) dominates the motion inside its Roche lobe, so particle orbits there are circling around the planet, not the star. The circumstellar orbits in the vicinity of the planet’s orbit are affected, too. Bodies on “disk orbits” (meaning the disk of bodies circling around the star) have Jacobi constants C depending on the orbital separation parameter x = (r-a)/a (r=initial circular orbit radius far from the planet, a = planet’s orbital radius). If |x| is large enough, the disk orbits are forbidden from approaching L 1 and L 2 and entering the Roche lobe by the energy constraint. Their effective energy is not enough to pass through the saddle point of the effective potential. Therefore, disk regions farther away than some minimum separation |x| (assuming circular initial orbits) are guaranteed to be Hill-stable, which means they are isolated from the planet. C CLCL

CLCL C Hill stability of circumstellar motion near the planet On a circular orbit with x = (r-a)/a, At the L 1 and L 2 points Therefore, the Hill stability criterion C(x)=C L reads or Example: What is the extent of Hill-unstable region around Jupiter? Since Jupiter is at a=5.2 AU, the outermost Hill-stable circular orbit is at r = a - xa = a a = 3.95 AU. Asteroid belt objects are indeed found at r < ~4 AU (Thule group at ~4 AU is the outermost large group of asteroids except for Trojan and Greek asteroids)

Back to the formation scenario: Isolation, Giant Impacts

Stopping the runaway growth of planetary cores Roche lobe radius grows non-linearly with the mass of the planet, slowing down the growth as the mass (ratio) increases. The Roche lobe radius r L is connected with the size of the Hill-stable disk region via a factor 2*sqrt(3), which, like the size of r L, we derived already in this lecture. This will allow us to perform a thought experiment and compute the maximum mass to which a planet grows spontaneously by destabilizing further regions.

Isolation mass in different parts of the Minimum Solar Nebula * Based on Minimum Solar Nebula (Hayashi nebula) = a disk of just enough gas to contain the same amount of condensable dust as the current planets; total mass ~ 0.02 M sun, mass within 5AU ~0.002 M sun Conclusions: (1) the inner & outer Solar System are different: critical core=10M E could only be achieved in the outer sol.sys. (2) there was an epoch of giant impacts onto protoplanets when all those semi-isolated ‘oligarchs’ where colliding.