Before Deuterium Ignition

Gas accreting onto the protostar generates energy as follows

$$L_{\rm acc} =\frac{GM_*\dot{M_*}}{R_*}$$ (4.117)

where $M_*$, $R_*$ and $\dot{M_*}$ represent the mass and radius of protostar and the mass accretion rate.

The central temperature increases with time. Finally thermonuclear fusion reaction of Deuterium $^2{\rm H}(p,\gamma)^3{\rm He}$ begins. Before Deuterium burning begins, the protostar is radiative, that is, the energy is transported radiatively.

When a fresh gas of $\Delta m$ accrets, thermal energy of $\Delta U=GM_*\Delta m/R_*$ increases. Virial theorem (eq.[2.120]) requires the potential energy must decrease (increase in the absolute volume) at

$$\Delta W = - 2 \Delta U\simeq -\frac{GM_*\Delta m}{R_*/2},$$ (4.118)

to achive a mechanical equilibrium. As a result, the fresh gas must contract to about half the radius at which the gas first joined the core (Stahler, Shu, & Taam 1980).

When the free-falling fresh gas accrets on the static star, an accretion shock forms. Since the gas temperature is increased with passing the shock front, temperature of the postshock gas, $T_g$ , is much higher than that of the radiation, $T_r$. Thus, the postshock gas cools very effectively. The postshock region with $T_g > T_r$ is called radiative relaxation region. The outgoing luminosity at the accretion shock is much larger than that inside of the radiative relaxation region.

In the case of low mass stars since the Kelvin-Helmholtz contraction time $t_{K-H}\simeq GM_*^2/R_*L$ ($L$ represent the luminosity at the base of the radiative relaxation region) is much longer than the acctretion time scale $t_{acc}\simeq M_*/\dot{M}_*$. This gives $L_{\rm acc}\simeq {GM_*\dot{M_*}}/{R_*}$ is much larger than $L$, which is consistent with the above statement that in the radiative relaxation region a large amount of accretion luminosity is radiated away. Since $t_{K-H} \gg t_{acc}$, the specific entropy inside the relaxation region is essentially frozen to that when the gas obtained passing through the relaxtaion region. Figure 4.21 taken from Stahler, Shu & Taam (1982) shows the distribution of the specific entropy against the accumulated mass $M_r$. The bottom curve corresponds to the state before the nuclear burning begins when no entropy generation occurs. The temperature increases with mass, bacause the star must be compressed to support an extra mass. After the temperature becomes high enough for Deuterium burning reaction $^2{\rm H}(p,\gamma)^3{\rm He}$ as $T\sim 10^6{\rm K}$, an extra energy is liberated by the nuclear fusion reaction. This increases the specific entropy mainly in the offcenter region. Figure 4.21 clearly shows that shell Deuterium burning occurs at $M\sim 0.025M_\odot$.

Figure 4.21: Specific entropy distribution is plotted against the accumulated mass (Stahler, Shu, & Taam 1982). The bottom curve shows the distribution at $t=2.3\times 10^4{\rm yr}$ ( $M_*=0.23M_\odot$). The top curve is for $t=3.4\times 10^4{\rm yr}$. The dashed curve indicates the entropy distribution if no convection is generated. While, the solid curves represent that of new distribution achieved with convection.