From the original burst, fraction of stellar mass still on the Main Sequence

[fab13]

<Moderator's note: Moved from a technical forum and thus no template.>

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

 

[phyzguy]

You need to know how long a star lives on the main sequence as a function of stellar mass. Do you have an expression for this? More massive stars have shorter lifetimes and leave the main sequence sooner, so after 10 billion years only the lower mass stars will still be on the main sequence.

 

[fab13]

You need to know how long a star lives on the main sequence as a function of stellar mass. Do you have an expression for this? More massive stars have shorter lifetimes and leave the main sequence sooner, so after 10 billion years only the lower mass stars will still be on the main sequence.

I don't know an example of this kind of law : could you give me one of them (if they are all different) ? I would need to have time of main sequence as a function of star mass.

thanks

 

[phyzguy]

Is this homework? If so, surely you have discussed stellar evolution. Did you try Googling 'Stellar lifetime"?

 

[fab13]

I have found the following relation between Luminosity and Mass :

$L=L_{\odot}\bigg(\dfrac{M}{M_\odot}\bigg)^{3/5}$

How to integrate this relation into Salpeter IMF : $\Phi(m)\text{d}m=\Phi_{0}\,m^{-2.35}\,\text{d}m$

??

 

[phyzguy]

I have found the following relation between Luminosity and Mass :

$L=L_{\odot}\bigg(\dfrac{M}{M_\odot}\bigg)^{3/5}$

You should check this again. The mass-luminosity relation is much steeper than this. If you know how the mass-luminosity relationship, how would you use this to find the relationship between mass and lifetime? How much nuclear fuel does the star have as a function of mass?

 

[fab13]

From http://hyperphysics.phy-astr.gsu.edu/hbase/Astro/startime.html#c1 , I have found the lifetime for s star :

$\tau = 10^{10}\,\bigg(\dfrac{M}{M_{\odot}}\bigg)^{2.5}\,\text{years}$

If I take $M=M_{\odot}$, I get $\tau = 10^{10}\,\text{years}$

But in my case , it is assumed that I have a burst 10 bliions ago and now that red-giants dominate the luminosity in B-Band : from these elements, I have to find an estimation of the total stellar mass of this galaxy.

Have you got tracks or suggestions to get it ?

 

[phyzguy]

Now we're getting someplace. So any star greater than Msun will no longer be on the main sequence. in your original post, you said you were supposed to calculate, "the fraction of stellar mass still surviving as stars in the main sequence". Do you see how to do that now?

 

[fab13]

the calculation about thefraction of stellar mass still surviving as stars in the main sequence, has been done :

like this :

$\begin{equation} \eta = \dfrac{\int_{m_{min,10 Gyr}}^{m_{max,10 Gyr}} m\Phi(m)\, dm}{\int_{0.1}^{120} m\Phi(m)\, dm}\end{equation}$

with $m_{min,10Gyr}=0.1 \text{M}_{\odot}$ and $m_{max,10\,Gyr}=1 \text{M}_{\odot}$

the result is equal to $\eta=60\%$ of stars that survives into Main Sequence.

Now, I would like to compute the mass of this galaxy by using Salpeter relation and luminosity of red-giants which dominates the band-B luminosity filter.

Do you see how I could perform this calculation of galaxy mass ?

 

[phyzguy]

Note that it didn't ask for the fraction of stars, it asked for the fraction of mass.

No, I don't know how to calculate your new question concerning red giants. What does the question say, exactly?

 

[fab13]

ok, the first question (where I get 60% of fraction mass that survives on Main sequence), it is :

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

Now, the other question, that I didn't solve for the moment is :

If the luminosity in the B band is dominated by stars of in the
RG branch, with masses m 1M⊙ (within 10%) and average luminosities
1000L⊙, estimate the total stellar mass of this galaxy using the same assump-
tions as in previous question regarding the IMF.

Nothe that one gives the absolute magnitude in B-band filter : $M_{B} = -21.22$

 

[phyzguy]

OK, so given the absolute magnitude in the B band, you should be able to calculate the total luminosity in the B band. Then calculate how many stars with a luminosity of 1000 Lsun it takes to give that luminosity. Then that tells you how many stars are between 0.9 Msun and 1.1Msun. So from that you can get $\Phi_0$. Then from that you can calculate the total mass of the galaxy.

 

[fab13]

the calculation about thefraction of stellar mass still surviving as stars in the main sequence, has been done :

like this :

$\begin{equation} \eta = \dfrac{\int_{m_{min,10 Gyr}}^{m_{max,10 Gyr}} m\Phi(m)\, dm}{\int_{0.1}^{120} m\Phi(m)\, dm}\end{equation}$

with $m_{min,10Gyr}=0.1 \text{M}_{\odot}$ and $m_{max,10\,Gyr}=1 \text{M}_{\odot}$

the result is equal to $\eta=60\%$ of stars that survives into Main Sequence.

Now, I would like to compute the mass of this galaxy by using Salpeter relation and luminosity of red-giants which dominates the band-B luminosity filter.

Do you see how I could perform this calculation of galaxy mass ?

A little remark : I think that $\eta$ represents the fraction but towards the total mass and not the total number of stars.

 

[fab13]

Hello, @phyzguy,

can I send you with private message my homework that I have finished ? , you will be able to see the reasoning, and tell me if my calculations are correct .
Regards

 

[phyzguy]

Hello, @phyzguy,

can I send you with private message my homework that I have finished ? , you will be able to see the reasoning, and tell me if my calculations are correct .
Regards

I guess that's OK, but why not share it here publicly so others can benefit and comment as well?