Hydrogen atoms fuse to form helium through the proton-proton chain which fuses four protons into one alpha particle (nucleus of ${}^{4}He$) and releases two neutrinos, two positrons and energy in the form of gamma photons. Although photons travel at the speed of light, the random motions they experienced inside the sun takes them thousand of years to leave the Sun' center. This random motion is due to the dense plasma in the Sun's interior since each photon permanently collides with an electron and gets deviated from its original path. The Energy released by fusion moves outward up to the top of the radiation zone, where the temperature drops to about 2 million K, then the photons get absorbed by the plasma more easily and this creates the necessary conditions for convection. This creates the convection zone of the zone. Then the plasma rises and the photons are carried to the photosphere where the density of the gas is low enough that they can escape. They mostly escape as visible photons, as their initial energy is lost through the random motion in the radiactive zone, and the absortion in the convective zone.
Let's assume the light from the Sun is parallel, then the shadow of Earth looks like this:
The dotted line is the orbit of the satellite at a height $h$ (I've exaggerated the height a bit to make the diagram clearer). All we have to do is calculate the angle $\theta$, because the time the satellite is in the Earth's shadow is simply:
$$ t = \tau \frac{2\theta}{2\pi} \tag{1} $$
where $\tau$ is the period of the satellite. It should be obvious from the diagram that the distance I've labelled as $d$ is equal to the radius of the Earth, $r$, and therefore:
$$ (r + h) \sin\theta = r $$
or:
$$ \theta = \arcsin \left( \frac{r}{r + h} \right) \tag{2} $$
Finally the period of the satellite, $\tau$, is given by:
$$ \tau = 2\pi\sqrt{\frac{(r+h)^3}{GM}} \tag{3} $$
where $M$ is the mass of the Earth and $G$ is Newton's constant.
Putting all this together, for a satellite at 1000km equation 2 gives us the angle $1.044$ radians (59.8°), and equation 3 gives us the period $\tau = 105.15$ minutes. Feeding these results into equation 1 tells us that the time the satellite is in the Earth's shadow is $34.9$ minutes.
Best Answer
Light takes a long time to escape not because the sun is particularly large but because they run into a lot of electrons and protons on the way. So the light is taking a path that is more like a drunkard's walk than a straight line. With neutrinos, though, they interact very rarely. A typical neutrino can pass through more than a light-year of lead before it runs into anything. The only reason we detect any neutrinos at all is because there are so many of them that even events that are rare happen a few times.