Jeans Mass

Gravitational collapse well understood from time of Newtonian theory

Sir James Jeans first developed from this to a theory for gravitational collapse of a static, non-viscous fluid

Despite using Newtonian gravity, it’s a good approx.

Overdense regions

• Imagine a smooth distribution of matter containing a slightly denser-than-average region

• WHAT HAPPENS?

• Answer depends on relative strength of two forces– gravity

– fluid pressure

Overdense regions

• Recall hydrostatic equilibrium in stars

• If density of the region is not too high then pressure will support it and it will oscillate like a sound wave

(sound waves represent traveling pressure perturbations in a gas/fluid)

JEANS MASS

• If the density of the clump is very high then the pressure force cannot overcome gravity and the dense region collapses in on itself

There is a minimum mass required for the dense region to collapse in on itself (recall Chandresekhar limit) -this minimum mass is called the JEANS MASS

THE JEANS MASS AND THE ONSET OF COLLAPSE

• Summary:– Imagine an overdense

clump of mass M.– If clump is very small,

sound waves will smooth it out.

– If clump is very big, it will gravitationally collapse.

– The threshold mass separating these behaviors is called the Jeans Mass, MJ

JEANS MASS

ρ3

3

G

speed) sound(=JM

MJ obtained by equating gravitational energy of the perturbation (which controls its collapse) with the thermal energy (generates pressure to oppose collapse)

ρ is the density of the clump

THE JEANS MASS AND THE ONSET OF COLLAPSE

ρ3

3

G

speed) sound(=JM

MJ depends mainly on ratio of time a sound wave takes to cross a perturbation, compared to the time for gravitational collapse of the perturbation

THE JEANS MASS AND THE ONSET OF COLLAPSE

ρ3

3

G

speed) sound(=JM

If sound can cross the region before collapse can get underway then pressure waves can restore hydrostatic equilibrium

If collapse begins before sounds gets across the region then gravity wins

JEANS MASS

ρ3

3

G

speed) sound(=JM

So, gravitational collapse favors larger clumps containing more mass and/or low temperatures so sound travels slowly

Jeans Mass

• Explains how we got from the almost perfect homogeneity just after the big bang to the “lumpy” and “bumpy” situation we have nowadays?

• Basic answer – gravitational collapse of regions with mass > Jeans mass

JEANS MASS - more detailThe Virial Theorem states that, for a stable, self-gravitating, spherical distribution of equal mass objects (stars, galaxies, etc), the total kinetic energy of the objects is equal to minus 1/2 times the total gravitational potential energy.

so

the Jeans Condition

JEANS MASS - more detail

the Jeans Condition

there is a min mass below which thermal pressure prevents grav collapse

Now EG=-GMm/r =-GM2/r

N is # atoms, mean molecular weight, mp is mass of proton

JEANS MASS - more detail

combining these gives

high density favors collapse, high temperatures favor large Jeans Mass

JEANS MASS - more detail

In early Universe (at a time called the decoupling time) when T~ 3000 K, n ~ 6 x 103 -> MJ ~ 105 M

In the interstellar medium T~1000, n~ 103 -> MJ~ 500 M

So in the early universe the smallest thing which could collapse was massive ! 105 M

Today, smallish molecular clouds can collapse - form individual stars.

Jeans mass versus time/conditions for the early universe is an important constraint on how galaxies formed!