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In a previous question I learned that in each second only a miniscule portion of the total hydrogen in the Sun is converted to helium and that the number is 1/10^18 of its mass converted each second. Since it takes billions of years to run out of hydrogen what is the event(s) that causes the Sun to eventually stop using the hydrogen for the nuclear reaction? (and about how long for this to happen.)

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The core of the Sun is radiative. That means that energy is transported outwards primarily by photons and is stable to convection. This means, to first-order, that the centre of the Sun is not mixed up by convection.

As the hydrogen in the core is burned, it forms helium, which has a larger atomic mass. The helium sinks towards the middle and the core fills with "ash". The parts of the Sun well away from the core still have plenty of unburned hydrogen, but because there is no mixing it cannot get to the centre where it is hot enough to sustain hydrogen burning. As the hydrogen burning continues, the helium content of the core grows and the mean mass per particle increases. This causes the core to contract and get a little hotter to maintain pressure, and increases the rate at which the remaining core hydrogen is burned.

However, as the supply of fresh hydrogen peters out (in about 5 billion years time according to Pietrinferni et al. 2004 or 5.5 billion years accordng to Schroder & Connon Smith 2008) the contraction speeds up markedly. The helium core becomes inert (temperatures are too low yet to initiate helium burning), but lots of fresh hydrogen is brought into a shell-like region that can burn hydrogen and it is this that marks the beginning of the growth of the Sun towards a red giant.

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In about four billion years the sun will have used up most of the hydrogen in its core and fuse helium into heavier elements. The fusion of helium releases more energy than the fusion of hydrogen. This will cause the sun to expand to, at least, the orbit of Venus and likely the Earth. Both planets will, of course, be vaporized.

There was a similar question on SE answered here.

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    $\begingroup$ Timescale and sequence of events is incorrect. Shell H-burning commences smoothly after core H-burning ends and before core He-burning begins. The Sun expands to a red giant before core He-burning begins. $\endgroup$ – Rob Jeffries Aug 19 '15 at 9:26

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