A very slow journey with a fast ending
The Last Eight Minutes Were the Easy Part
Sunlight crosses from the Sun to your face in eight minutes. Getting out of the Sun took very much longer.
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Step 1 of 5
The energy is made right in the middle.
Fusion only happens in the core, where the crush is hardest — the inner quarter or so of the radius. Everything outside that is the Sun's problem of getting the energy out, and the Sun is very bad at it on purpose.
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Step 2 of 5
Outside the core, the Sun is packed absurdly tight.
The material in the radiative zone is denser than lead in places, and it is opaque. Energy released as a burst of light travels a tiny distance, gets absorbed by a particle, and is re-emitted a moment later in a completely random direction.
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Step 3 of 5
Random direction means it makes almost no progress.
This is a stagger, not a walk. Take a step, spin round, take another. You do drift outward eventually, but slowly, because half the time you head back the way you came. Physicists call it a random walk and it is spectacularly inefficient.
Strictly, no single particle of light survives the trip. Each one is absorbed and a new one emitted. What travels out is the energy, not the messenger.
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Step 4 of 5
How long it takes is honestly uncertain.
Each mark stands for a slice of the estimate. The last group is the part everybody quotes. You will see 100,000 years quoted, and also 10,000, and also 170,000. The answer depends on how you model the density and opacity at every depth, and on whether you are asking about energy or about a hypothetical single photon. Published estimates spread over more than a factor of ten.
It is a good example of a number that is famous, roughly right and much less settled than its fame suggests.
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Step 5 of 5
Then it crosses 150 million kilometres in eight minutes flat.
The short leg is the one that takes no time. Once it leaves the surface, there is nothing in the way. Empty space does not absorb, does not scatter and does not slow anything down. The hardest part of the journey was the first 700,000 kilometres, not the last 150 million.
The short version
Energy from the Sun's core staggers outward through dense plasma for a very long and not-precisely-known time, and only the final 150 million kilometres of empty space takes the famous eight minutes.
Try it yourself
Take a step, spin to face a random direction, step again — twenty times. Measure how far you actually got from where you started. It will be far less than twenty steps' worth. That is why the Sun is so slow to leak.