Astronomy with an online telescope
Astronomy with an online telescope

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Astronomy with an online telescope

3.3 Estimating the Sun’s lifetime

In the last section, you saw that nuclear reactions in the core of the Sun result in the conversion of 4.3 million tonnes of matter into energy every second. While this may seem like a staggering amount, it should be taken in the context of the enormous mass of the Sun itself – there is no danger of the Sun running out of nuclear fuel any time soon. In fact, we can use this rate of consumption together with our knowledge of the structure and composition of the Sun to make an estimate of how long the supply of hydrogen in the Sun will last.

Activity 4 Estimating the lifetime of the Sun

Timing: Allow approximately 10 minutes

In this optional activity, if you are familiar with working with large numbers expressed in scientific notation, you can try working out the lifetime of the Sun based on the following facts and figures.

If you are not confident with the calculation, you can still follow the chain of reasoning and then click to reveal the answer below.

First, you can calculate the amount of mass available in the core of the Sun that can be converted into energy. You can do this by starting with the total mass of the Sun and narrowing it down as follows:

  • The total mass of the Sun is 2.00 times prefix multiplication of 10 super 30 times k times g
  • 75% of this mass is hydrogen
  • 12.5% of this is in the core and is able to take part in nuclear fusion
  • only 0.73% of this mass is released as energy in the ppI chain reaction

This gives a mass 1.37 times prefix multiplication of 10 super 27 postfix times k times g postfix times that would have been available for conversion into energy at the start of the Sun’s life.

Although the Sun’s luminosity (energy output) has actually varied slightly during its lifetime, for the purposes of estimating how long this supply of mass for conversion into energy will last, it is reasonable to use the present value of 4.3 million tonnes (4.3 multiplication 10 super nine postfix times k times g postfix times) per second as a constant average value.

Remembering that there are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day and 365.25 days in a year, calculate the lifetime of the Sun by working out how long (in years) the available mass of 1.37 times prefix multiplication of 10 super 27 postfix times k times g postfix times would last at this rate of consumption.

Answer

First, we find the number of seconds that the nuclear fuel will last by dividing the available mass by the mass used per second:

multiline equation line 1 equation sequence part 1 Lifetime equals part 2 Total available mass divided by Mass used per second equals part 3 1.37 multiplication 10 super 27 postfix times k times g divided by 4.3 multiplication 10 super nine postfix times kg equals part 4 3.19 multiplication 10 super 17 times seconds line 2 and then convert this into years by dividing by the number of seconds in a year line 3 equation sequence part 1 Lifetime equals part 2 3.19 multiplication 10 super 17 postfix times seconds divided by 365.25 multiplication 24 multiplication 60 multiplication 60 postfix times seconds in a Year equals part 3 1.01 multiplication 10 super 10 times years

Our estimate of the lifetime of the Sun is therefore 10 billion years. Since we know that the Sun is approximately 4.6 billion years old, the Sun is currently just under halfway through its stable lifetime. There is plenty of time left for the Earth and the Solar System.

AOT_1

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