# Finding the lifetime of a main sequence star.

• constantinou1
In summary, the main sequence lifetime of the Sun is approximately 10 billion years, assuming that only the core mass of the star is converted from hydrogen to helium. This accounts for the fact that the entire mass of the star does not have to be converted for core fusion to cease.
constantinou1

## Homework Statement

Given the following data, calculate the main sequence lifetime of the Sun (in years), assuming that all the initial mass is hydrogen and all of it is converted into helium.
Mass of the Sun = M = 2x1030kg
Luminosity of the Sun = L = 4x1032W
Energy released in the proton-proton reaction (4H $\rightarrow He$) is Efusion = 4x10-12J
Comparing the mass of the final helium-4 atom with the masses of the four protons reveals that 0.7% of the mass of the original protons has been lost.

## Homework Equations

L = $\frac{E}{t}$

where L is the stars Luminosity, E is the total energy supplied by hydrogen fusion and t is the stars main-sequence lifetime.

The main-sequence lifetime of a star is also given by:

t = $\frac{fMc^2}{L}$

where f is fraction of of the stars mass converted into energy and c being the speed of light.

## The Attempt at a Solution

- First attempt...

Using the data given, I can just punch it into the equation above, which gave me:

t = [0.007*2x1030*(3x108)2] ÷ 4x1026

This gives me an answer of 3.15x1018 seconds.

Converting this into years yields...

3.15x1018 x $\frac{1}{60*60*24*365}$

which gives 99.9x109 years.

This means the main sequence lifetime of the Sun is about 100 billion years, which can't be correct because in almost every textbook I look in, it states its main sequence lifetime should be 10 billion years. I know that I am by a factor of 10 out somewhere, but I can't seem to find where I have gone wrong.

- Second attempt...
The amount of fusion reactions that take place in the Sun in order to provide the necessary Luminosity is...

L ÷ Efusion = (4*1026) ÷ (4x10-12) = 1038 s-1

Using the above, I can find the total mass of hydrogen converted into helium per unit time (using Wikipedia to find the mass of hydrogen nuclei)...

R = rate at which hydrogen is converted = 4(1.67x10^-27)(10^38) ≈ 670x10^9 kg s-1

I multiplied by 4 in the above because 4 hydrogen nuclei are required for 1 fusion reaction.

The lifetime of the Sun can now be found by dividing the mass of the Sun by the rate at which the hydrogen is converted to helium, so...

$\frac{M}{R}$ = 2x1030 ÷ 670x109 ≈ 3x1018s.

Converting this into years yields...

3.1x1018 x $\frac{1}{60*60*24*365}$

which gives 99.9x109 years.

This is the exact same answer that I got in the first attempt! Either what I have done is correct or I have gone wrong in the two attempts somewhere, but I know that I must be going wrong somewhere because the answer is out by a factor of 10 and its quite frustrating now.

Thanks in advance for any replies and sorry for the bad layout of the 'attempt at a solution' section.

Welcome to PF constantinou1!

constantinou1 said:

## Homework Statement

Given the following data, calculate the main sequence lifetime of the Sun (in years), assuming that all the initial mass is hydrogen and all of it is converted into helium.
Mass of the Sun = M = 2x1030kg
Luminosity of the Sun = L = 4x1032W
Energy released in the proton-proton reaction (4H $\rightarrow He$) is Efusion = 4x10-12J
Comparing the mass of the final helium-4 atom with the masses of the four protons reveals that 0.7% of the mass of the original protons has been lost.

## Homework Equations

L = $\frac{E}{t}$

where L is the stars Luminosity, E is the total energy supplied by hydrogen fusion and t is the stars main-sequence lifetime.

The main-sequence lifetime of a star is also given by:

t = $\frac{fMc^2}{L}$

where f is fraction of of the stars mass converted into energy and c being the speed of light.

## The Attempt at a Solution

- First attempt...

Using the data given, I can just punch it into the equation above, which gave me:

t = [0.007*2x1030*(3x108)2] ÷ 4x1026

This gives me an answer of 3.15x1018 seconds.

Converting this into years yields...

3.15x1018 x $\frac{1}{60*60*24*365}$

which gives 99.9x109 years.

This means the main sequence lifetime of the Sun is about 100 billion years, which can't be correct because in almost every textbook I look in, it states its main sequence lifetime should be 10 billion years. I know that I am by a factor of 10 out somewhere, but I can't seem to find where I have gone wrong.

There is nothing wrong with your arithmetic, but rather with your astrophysics. You are assuming that the main sequence lifetime will end when ALL of the mass of the star is converted from hydrogen to helium. However, this is not necessary. In practice, it is sufficient for the mass in the CORE of the star (the only place that is hot enough for fusion to be occurring in the first place) to be entirely converted from hydrogen to helium. At this point, core fusion ceases, and hence the star goes off the main sequence. So, your value for "M" here would have to be the core mass, rather than the total stellar mass. I'm not sure how much of the mass of the sun is in its core, but presumably the fraction is ~1/10.

EDIT: It looks like your second solution method has the same problem.

Ahh ok, I understand now, thanks a lot cepheid for making that clear. Taking that into consideration, the answer that I am coming to now seems to make a lot more sense.

## What is the main sequence in a star's lifetime?

The main sequence is the stage in a star's lifetime where it fuses hydrogen atoms into helium in its core, providing the energy that allows it to shine. This stage typically lasts for billions of years.

## How do scientists determine the lifetime of a main sequence star?

Scientists use a combination of observational data and theoretical models to determine the lifetime of a main sequence star. They can measure a star's luminosity and mass, and use these values to predict its lifespan based on our understanding of stellar evolution.

## What factors affect the lifetime of a main sequence star?

The main factors that affect a star's lifetime are its initial mass and its composition. A more massive star will burn through its fuel faster and have a shorter main sequence lifetime. A star's composition also plays a role, as a higher abundance of heavier elements can affect the fusion process and alter its lifespan.

## Can the lifetime of a main sequence star be changed?

Yes, the lifetime of a main sequence star can be affected by external factors such as interactions with other stars or the presence of a binary companion. In some cases, these interactions can cause a star to prematurely evolve off the main sequence.

## What happens to a main sequence star after its lifetime ends?

Once a main sequence star runs out of hydrogen fuel in its core, it will begin to evolve into a red giant or supergiant. This marks the end of its main sequence lifetime and the beginning of a new stage in its evolution.

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