The Formation of Fragments at Corotation in Isothermal Protoplanetary Disks

Abstract

Numerical hydrodynamics simulations have established that disks which are evolved under the condition of local isothermality will fragment into small dense clumps due to gravitational instabilities when the Toomre stability parameter $Q$ is sufficiently low. Because fragmentation through disk instability has been suggested as a gas giant planet formation mechanism, it is important to understand the physics underlying this process as thoroughly as possible. In this paper, we offer analytic arguments for why, at low $Q$, fragments are most likely to form first at the corotation radii of growing spiral modes, and we support these arguments with results from 3D hydrodynamics simulations. Comment: 21 pages, 1 figure

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arXiv:0709.1445v1 [astro-ph] 10 Sep 2007
The Formation of Fragments at Corotation in Isothermal
Protoplanetary Disks
Short Title: Disk FragmentationArticle Type: Journal
Richard H. Durisen
Astronomy Department, Indiana University, Bloomington, IN 47405
durisen@astro.indiana.edu
Thomas W. Hartquist
School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK
twh@ast.leeds.ac.uk
and
Megan K. Pickett
Department of Physics, Lawrence University, Box 599, Appleton, WI 54912
megan.pickett@lawrence.edu
Received
; accepted

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ABSTRACT
Numerical hydrodynamics simulations have established that disks which are
evolved under the condition of local isothermality will fragment into small dense
clumps due to gravitational instabilities when the Toomre stability parameter Q
is sufficiently low. Because fragmentation through disk instability has been sug-
gested as a gas giant planet formation mechanism, it is important to understand
the physics underlying this process as thoroughly as possible. In this paper, we
offer analytic arguments for why, at low Q, fragments are most likely to form first
at the corotation radii of growing spiral modes, and we support these arguments
with results from 3D hydrodynamics simulations.
Subject headings: accretion, accretion disks — hydrodynamics — instabilities —
planetary systems: formation — planetary systems: protoplanetary disks

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1. Introduction
The idea that gas giant planets can form rapidly through gravitational instabilities
(GIs) in protoplanetary disks (Boss 1997, 1998), usually called the “disk instability” theory,
has now been subjected to intensive study with numerical hydrodynamics techniques
(Pickett et al. 1998, 2000, 2003; Nelson et al. 1998, 2000; Nelson 2006; Boss 2000, 2001,
2002, 2003, 2005, 2007; Gammie 2001; Mayer et al. 2002, 2004, 2007; Johnson & Gammie
2003; Rice et al. 2003, 2005; Mej´ ıa et al. 2005; Cai et al. 2006; Boley et al. 2006, 2007).
Although all investigators agree that massive, cold protoplanetary disks fragment into
dense clumps under conditions of rapid cooling or local isothermality, there is not universal
agreement that these fragments are sufficiently long-lived to be considered bound gas giant
protoplanets or that fragmentation will occur under realistic disk conditions (Durisen et al.
2003, 2007; Boss 2004, 2005; Rafikov 2005, 2006; Pickett & Durisen 2007a). This important
question may ultimately be decided through simulations alone, but we will gain greater
confidence in the numerical results if we understand the physics of fragmentation and clump
longevity through analytic arguments, even if only approximate.
As a first step in this direction, we consider the special case of a disk evolved under a
“locally isothermal” assumption, i.e., the disk is assumed to maintain its initial temperature
at all positions (Boss 1998; Pickett et al. 1998; Nelson et al. 1998). For brevity, most
GI researchers refer to this as an “isothermal” disk, even though the temperature of the
disk is not the same at all positions. Local, thin-disk simulations with a variety of Qs by
Johnson & Gammie (2003) suggest that fragmentation of isothermal disks occurs for Q <
about 1.4, and the set of all global 3D hydrodynamics simulations for isothermal disks
referenced in the preceding paragraph generally supports the occurrence of fragmentation
for such low Qs. The same set of simulations, taken together, also generally confirms that
disks are subject to growth of nonaxisymmetric GIs for Q < about 1.7 or so and are stable

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for higher Qs. In simulated disks that exhibit GIs, the instabilities are initiated by the
growth from noise of discrete spiral modes (Pickett et al. 1998, 2000, 2003). Our own latest
isothermal disk simulations, which will be reported in detail elsewhere (Pickett & Durisen
2007b), show three types of behavior: 1) at low Q, prompt fragmentation of spiral GI
modes as they first become nonlinear, 2) at intermediate Q, delayed fragmentation in the
nonlinear regime, probably due to nonlinear interactions of multiple spiral modes, and 3) at
the highest unstable Qs, nonlinear spirals which do not fragment.
In this paper, we explore the idea that prompt fragmentation in isothermal disks
tends to occur near the radius at which a discrete growing spiral mode corotates with the
disk gas (the “corotation radius” or CR). In Section 2, we give simple analytic arguments
that compression by spiral shocks must not be too large if the shock-compressed gas is
going to be able to fragment and that the smallest compression in a spiral mode should
be found near its CR. In Section 3, we describe representative results from a simulation in
Pickett & Durisen (2007b) where prompt fragmentation does in fact occur in the vicinity
of the CR for a discrete fast-growing mode at low Q. The simulation allows us to estimate
some of the uncertain factors in the analysis of Section 2 and verify that the analytic
arguments do apply. Section 4 summarizes our main conclusions.
2. Compression-Induced Stability Against Fragmentation
Consider cylindrical coordinates (r,φ,z) with the rotation axis of the disk along the
z-axis, the disk midplane at z = 0, and the sense of rotation in the positive φ-direction.
In the absence of spiral structures and shocks, a low-Q disk has a half-thickness H ≪ r in
the z-direction defined such that Σ1= 2ρ1H, where Σ1is the surface density and ρ1is the
midplane density. We shall assume that the initially fastest-growing discrete spiral mode
develops more rapidly than any other instability which may give rise to fragments. A spiral

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shock at r associated with the mode then sweeps material into a sheet of vertical height H
with thickness L normal to the spiral shock front. Because the gas responds isothermally
to the shock, there is no vertical jump behind the shock, and the vertical height of the disk
remains essentially the same on both sides of the shock (Boley & Durisen 2006).
We will analyze the gravitational stability of the swept-up gas by assuming that it is
well approximated by a thin, plane-parallel sheet. This requires L ≪ r and
L < 2α1H,(1)
where α1is a constant of order unity which accounts for uncertainty about how much
smaller L must be than 2H for our subsequent analysis to be at least approximately valid.
Throughout, we will introduce similar factors, each of which is presumably of order unity
and designated by an αj, where j is an index. They account for uncertainties associated
with our assumptions.
If the density ρ2in the swept-up sheet can be considered roughly constant over scales
of order L, which seems reasonable, then potential energy per unit area of the gas sheet is
EG= −πGρ22L3/6, (2)
where G is the gravitational constant. There are several effects that can counter
the tendency of self-gravity to cause fragmentation, including thermal pressure, tidal
gravitational stresses, and shearing motions. The Virial Theorem suggests that an infinite,
isothermal, plane-parallel, ideal gas slab will be stable against fragmentation due to its
thermal energy alone if |EG| < 2ET, where ET is the total thermal kinetic energy of the
slab per unit area. This gives
πGρ2L2< 18α2ci2, (3)
where ciis the isothermal sound speed and α2accounts for the uncertainty due to deviations