Part1 - Theory of Gravitational Instability Flashcards
(32 cards)
principle of Jeans mass
how much mass can you have in a given volume of space before it starts to collapse under its own weight
we derive Jeans mass from first principles by
balancing thermal pressure against self gravity
two expressions for pressure used in Jeans mass derivation
1.P=ρ Kb T/ m from ideal gas law and V=Nm/ρ
- P=Gρ^2L^2 from P=Mg/A (g=GM/L^2)
we need to equate the two expressions for pressure to determine the
critical side length (Jeans length) above which the volume of gas will collapse under its own weight
the propagation of a pressure wave will cause
areas of compression (higher density) and rarefaction (lower density)
can quantify areas where density is too high in terms of the
sound speed of the pressure wave travelling through the gas
jeans length found by noting that
dP/dρ KbT/m = Cs^2
(Cs=sound speed)
if the column of gas has a length larger than Lj or a mass larger than Mj, then
gravitational collapse will begin and a ‘structure’ may form
we will answer ‘how do large-scale structures arise’ by
perturbing our simple Newtonian fluid model
linearisation of equations: AB=
(A0+A1)(B0+B1)
APPROX = A0B0 +A1B0 + A0B1
assuming A1B1 can be neglected as it is so small
linear approximation
only first order terms are kept - second-order terms are neglected
equations for a small perturbation in a non-expanding universe
fluid equations for an ideal gas and use an isotropic, homogeneous and non-expanding universe
a small perturbation will obey
ρ1 dot +ρ0∇ . v1 =0
a small perturbation will obey v1 dot=
-∇φ1 -1/ ρ0 ∇ P1
solution for a small perturbation - disturb the gas by
having a density wave pass through it
a plane wave is a ‘small enough’ disturbance and will
obey the principle of superposition
assume that the perturbation in density ρ1 has the form
of a plane wave disturbance
in this derivation, 0 refers to
equilibrium values
not present time values
if the characteristic length of the perturbation is larger than Jeans length, then
the Jeans instability criterion reveals that gravitational collapse will occur
if M is the mass associated with a gravitational perturbation, it will be
unstable and will fragment if M>MJ
the new temperatures T’ and densities ρ’ in the fragment will
define a new Jeans mass MJ’
If MJ’ is again exceeded, then
the fragment will break up again and the process continues
for a continuous process of breakup, we require
Jeans mass to be a decreasing function of density
The coefficients are now time-dependent. The consequence is that
we lose the exponential growth of instability that was obtained in a static universe in favour of a power law growth
