What percentage of the energy in a dust cloud must be lost before it can collapse into a star?

With reference to this previous question about how dust clouds can collapse to form stars:

How is hydrostatic pressure overcome when a star is formed?

The answer given is that they must radiate away some heat.

Of the total energy in a dust cloud, can someone give me an estimate of what %age of its energy must be radiated away as heat before a star can form?

• How big is the cloud? What are the initial conditions? Can't answer without that. – Rob Jeffries Nov 5 '15 at 12:41
• Only a fraction of the cloud will give star(s), so it's difficult to bound the part on which energy must be measured to answer. – Fabrice NEYRET Nov 5 '15 at 13:36
• I don't really know what the initial conditions are, just looking for a very approximate answer. physics.stackexchange.com/questions/167496/…, estimates 10^41 down to 10^35 joules for our sun. – user2800708 Nov 5 '15 at 14:42

You can get a rough idea from the virial theorem. This tells us that for a gravitationally bound system the kinetic energy $T$ and the potential energy $V$ are related by:

$$2T = -V$$

or obviously:

$$T = -\tfrac{1}{2}V$$

Suppose we start with our dust cloud particles at infinity with $T = V = 0$ and let the system collapse until the potential energy is $-V$, then conservation of energy tells us that the kinetic energy will be $+V$. But the virial theorem tells us that for a gravitationally bound system $T = \tfrac{1}{2}V$, so in order to be stable our dust cloud must have radiated away an energy of $\tfrac{1}{2}V$.

The point of all this is that the initial energy of the cloud that formed e.g. the Sun was approximately zero. It was cold, so it's kinetic energy was low, and it was very low density so its potential energy was negligable.

If we approximate the Sun as a uniform density sphere then its gravitational binding energy is given by:

$$V = \frac{3GM^2}{5R}$$

The radius $R$ would have been the radius at which ignition occurred and not the current radius, because after ignition the heat produced by fusion has made the Sun expand again. I can't find any confident statements of the Sun's radius at the point ignition occurred, but I have found suggestions that it was at about half its current radius, so let's guesstimate $R = 3.5 \times 10^8$ m. In that case the equation for the binding energy gives:

$$V \approx 4 \times 10^{41} \,\text{J}$$

So from the virial theorem we find the Sun radiated away about $2 \times 10^{41}$ J while it was forming.

• Obviously this is correct, but the question asks what fraction of the original energy in the dust cloud is radiated away. You are assuming that the initial energy is zero. Your answer is a restatement of the well known fact that half of the gravitational potential energy released is radiated away. – Rob Jeffries Nov 5 '15 at 19:51
• I guess the question is what must $V$ fall to before the star can ignite? I'll have a think about that. – John Rennie Nov 5 '15 at 20:46
• No that's not the question (and if it was, the answer is readily available). – Rob Jeffries Nov 5 '15 at 21:10