# From original burst, fraction of stellar mass still surviving on Main sequence

Suppose that all stars in this galaxy were born in a single major-merger burst event about 10 Gyr ago. From this original burst, I want to compute the fraction of stellar mass still surviving as stars in the main sequence ? For this, I have got to use a Salpeter IMF, and a star formation range between 0.1 and 120 solar masses.

What I have done is starting from Salpeter IMF : $$\Phi(m)\text{d}m=\Phi_{0}\,m^{-2.35}$$

with $$\Phi_{0}$$ a constant normalization.

From this, I integrate from $$m_{1}=0.1\,\text{M}_{\odot}$$ to $$m_{2}=120\,\text{M}_{\odot}$$

$$N(0.1

This result depends on the valeur of $$\Phi_{0}$$ and I don't know how to deal with it in order to get $$N(0.1 ?

Moreover, it seems that I have to take into account of the age of the major-merger burst event (10 Gyr).

From these 2 principles, how could I calculate the fraction of stars surviving in the main sequence ?

Any help is wlecome, Regards

• I would like to start a bounty on this post but unfortunately, the link doesn't appear, could anyone tell me why ? – youpilat13 Dec 16 '18 at 23:23

When you are calculating fractions, rather than absolute numbers, the value of $$\Phi_0$$ does not matter, since it will be a multiplying factor in both the numerator and denominator.
Finally, you were asked to find the fraction of stellar mass surviving, not the fraction of stars. The stellar mass existing between two mass intervals is $$M_* = \int_{m_1}^{m_2} m\Phi(m)\ dm$$
• -@Rob Jeffries. Thanks for your quick answer. So the fraction of stellar mass $\eta$ still surviving as stars in the main sequence is equal to : $$\eta = \dfrac{\int_{m_{min,\text{10\,Gyr}}}^{m_{max,\text{10\,Gyr}}} m\Phi(m)\, dm}{\int_{0.1}^{120} m\Phi(m)\, dm}$$ ? Is this the definition of mean mass towards the distribution $\Phi(m)$ ? What values can I take for $m_{min\,\text{10$\,$Gyr}}$ and $m_{max\,\text{10$\,$Gyr}}$ ? regards – youpilat13 Dec 16 '18 at 10:24
• -@Rob Jeffries. For the moment, I have taken for the lower limit integral : $m_{min,\text{10$\,$Gyr}}=0.1\,\text{M}_{\odot}$ and for upper limit : $m_{max,\text{10$\,$Gyr}}=1\,\text{M}_{\odot}$, Do you think that's a good approximation ? With these values, I get 60% of mass surviving into Main Sequence. – youpilat13 Dec 16 '18 at 16:08