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According to the virial theorem, when a protostar contracts, half of the gravitational potential energy is radiated and half is kept as kinetic energy of the falling material which in turn heats the star. But at some point the central temperature reaches $10^{6}$ degrees Kelvin and deuterium burning starts. Some sources mention that this deuterium burning lasts for about a million years and results in stopping further contraction during this million year, but once the deuterium is depleted, the contraction can continue.

Now, according to this, the total energy available by deuterium burning is comparable to that released by gravitational contraction. So is the deuterium burning energy responsible for heating the protostar ? Or is the energy released by this burning is only used to delay the contraction and so virial theorem is correct ?

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  • $\begingroup$ Since the core's temperature does not increase, then the energy from the deuterium fusion heats and expands the plasma around the core. It's possible that the rate of fusion generates enough energy to delay the contraction of the protostar. $\endgroup$ – LDC3 Dec 7 '14 at 3:00
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With the given information, several results are possible: All the kinetic energy is converted into heat and the required temperature for "burning" deuterium is not reached (no star). If the temperature is reached, then the energy released by the deuterium burning might be enough to keep the star from contracting any further. The star might expand a little if more deuterium than necessary is burned or contract some more if not enough deuterium is burned. If equilibrium is reached, and there is a reserve of deuterium, this equilibrium will continue until the deuterium is used up.

If kinetic energy remains, some of the deuterium energy is used to stop the contraction, some to maintain the attained size, and some to maintain the temperature of the core. Once the contraction is stopped, the deuterium is used to maintain the attained size, the core temperature, and replace the energy radiated.

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You should just think of Deuterium burning as a short-lived counterpart of main sequence hydrogen burning.

As the pre-main-sequence star (PMS star, or protostar if you like) contracts, its core reaches around $6\times 10^{5}$ K and D burning is smoothly initiated. The star is then held at roughly constant luminosity and radius, because if it contracted, the core would heat, the D burning would increase a lot and this would heat up the star and it would expand. And vice versa. Thus there is a quasi-equilibrium set up, that lasts as long as the D lasts.

All of the D gets burned up because the PMS star (or brown dwarf - D burning can last a lot longer in brown dwarfs) is fully convective and thoroughly mixed.

During D burning, since the contraction is almost halted then this is what must supply the vast majority of the thermal energy and luminosity of the star during that phase.

Have a look at the Figures and Tables in Burrows et al. (1997), which show evolutionary tracks for low-mass objects. Table 4 is especially helpful. This shows that D burning starts in a $0.04M_{\odot}$ brown dwarf after about 1 million years of contraction, after 2.7 million years it contributes 94% of the star's luminosity and halts after about 10 million years. Over the period 1 million to 10 million years, the radius only shrinks by about 15%. Yet if we work out the Kelvin-Helmholtz contraction timescale for the same object (ignoring the nuclear burning), which is given by $$ \tau_{KH} = \frac{3}{7} \frac{GM^2}{RL},$$ we find that $\tau_{KH}= 3$ million years. Thus the D burning succeeds in (almost) halting gravitational contraction.

Your question asks whether D burning heats the star/brown dwarf, or whether it delays the contraction and the virial theorem is correct. Well the answer is it does both and the virial theorem should always be applicable when the star is in a state of quasi-equilibrium.

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