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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 $$m_{2}=120\,\text{M}_{\odot}$$

$$N(0.1<m<120) = \int_{0.1}^{120}\,\Phi(m)\,\text{d}m = \Phi_{0}\,\bigg[\dfrac{0.1^{-1.35}-120^{-1.35}}{1.35}\bigg]$$

This result depends on the valeur of $$\Phi_{0}$$ and I don't know how to deal with it in order to get $$N(0.1<m<120)$$ ?

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

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  • $\begingroup$ I would like to start a bounty on this post but unfortunately, the link doesn't appear, could anyone tell me why ? $\endgroup$
    – user16492
    Dec 16, 2018 at 23:23

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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.

You have (almost, see below) successfully got an expression for the denominator of your fraction.

The numerator is found by calculating an equivalent integral from your lower limit to an upper limit that is instead defined by the most long-lived main sequence stars that are still "alive" - i.e. those with a lifetime equal to the age of our Galaxy.

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 $$M_* = \int_{m_1}^{m_2} m\Phi(m)\ dm$$

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  • $\begingroup$ -@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 : \begin{equation} \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}\end{equation} ? 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 $\endgroup$
    – user16492
    Dec 16, 2018 at 10:24
  • $\begingroup$ @youpilat13 To first order, yes. I think you should look at your notes on stellar evolution to see how lifetime depends on mass, but all the stars with 0.1 solar masses that were ever born are still alive. $\endgroup$
    – ProfRob
    Dec 16, 2018 at 15:34
  • $\begingroup$ -@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. $\endgroup$
    – user16492
    Dec 16, 2018 at 16:08

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