Calculating Main Sequence Lifetime.

[adwodon]

1. Calculate the main sequence lifetime in years of a 10-M_{[tex]\odot[/tex]} star if it has luminosity of 10⁴L_{[tex]\odot[/tex]} and 10% of its mass will be converted from hydrogen to helium in the core. What will the end state of this star be?

Ok so I wasn't really sure what to put for working here as I have no idea where to begin, I've been searching around for a forumla or something to calculate the lifetime but my lecturers notes are very vague and not helpful at all and numerous different google searches haven't helped in giving me a straight answer.

I found this formula somewhere but it doesn't make sense to me:

t_ms[tex]\approx[/tex][tex]\frac{M}{L}[/tex]

Its part of a larger question involving the Hertzsprung Russell diagram which I've done however this bit I am totally stumped, if any of you guys could point me in the right direction or explain it to me i'd really appreciate it.

Oh and I should point out this is from a past paper, the only relevant information I could imagine is given to help on the front of the paper is (excluding the standards like Planck / speed of light etc):

Solar mass: 2x10³⁰kg
Solar Bolometric Luminosity: 3.9x10²⁶W

 

[adwodon]

Ok so I used E=mc^2 and got the total energy of the converted material as 6x10^38J
Which would sustain that luminosity for 5 years...

Thats roughly the same answer I got from the equation in my original post but I thought it was wrong because 5 years seems waaaaay too short.

Is that a plausible answer?

 

[baba89]

Hi there,

I can understand your frustration with vague lecture notes and conflicting information online. However, let me try to explain the formula you found and how to calculate the main sequence lifetime of a star.

The formula you found, tms\approx\frac{M}{L}, is actually the correct one to use in this case. Let me break it down for you:

- tms stands for the main sequence lifetime, which is what we are trying to calculate.
- M stands for the mass of the star, which is given as 10 M\odot (10 times the mass of our Sun).
- L stands for the luminosity of the star, which is given as 104L\odot (104 times the luminosity of our Sun).

So, to calculate the main sequence lifetime, we simply need to divide the mass of the star by its luminosity. This gives us a result of 10 M\odot / 104L\odot = 10,000 years. This means that the star will spend approximately 10,000 years on the main sequence before evolving into a different stage.

Now, let's talk about the end state of this star. Based on the given information, we know that 10% of its mass will be converted from hydrogen to helium in the core. This means that the star will eventually run out of hydrogen fuel and start fusing helium in its core. It will then expand and become a red giant star. After this stage, the exact end state will depend on the mass of the star. If it is less than 8 M\odot, it will eventually become a white dwarf. If it is between 8-10 M\odot, it will become a neutron star. And if it is more than 10 M\odot, it will become a black hole.

I hope this explanation helps you understand how to calculate the main sequence lifetime and the end state of a star. If you have any further questions, feel free to ask. Keep up the good work in your studies!