This morning I was helping run a first-year test in a lecture room where, on the blackboards, was a discussion of the elementary mathematics involved in detecting a planet orbiting a star by the dimming of the star’s light.
I was struck by the statement: “Take the star to be a disc of uniform brightness.” This certainly agrees with our experience of looking at the sun and moon, and graphs in papers on planet detection seem to bear it out: the dimming is uniform as the planet transits the star.
Yet, if we look near the periphery of a glowing sphere, we see more surface area per pixel than if we look at the centre; if radiation were uniform in all directions, we would expect the limb of the sun or moon to be significantly brighter than the centre.
Does anyone have a simple explanation?
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I should say that there is no puzzle about why the full moon looks like a uniform disc. Near the limb, the surface illumination from the sun is lower (since the sun’s rays strike the surface at a very low angle) in exactly the proportion to compensate for the effect I mentioned.
I’m no astronomer, but I think the key is the (probably wrong) assumption that a star is a perfectly platonic evenly glowing sphere in a vacuum.
In reality there are all kinds of gasses and dust particles and crap around the star. The bright glow of the actual surface is more obscured by that stuff at the edges.
It’s like how in photos of the Earth you can’t see through the atmosphere as well at the edges of the disk, because you’re looking through more of it.
The surface is also textured with waves and ripples, it’s not perfectly smooth. That also throws a wrench in the theory that the radiation is evenly cast in every direction. That waviness will stop a decent amount of light from a shallow viewing angle. From a shallow angle each wave blocks the light from its own backside and the trough behind it.
There is actually limb darkening instead of brightening because you are looking through less of the stellar material. As the planet transits, it actually blocks more light when it passes through the center. Sure I see what you mean by more surface area, but the real question is if you look down your line of sight at all radiation coming down that line since it is indeed isotropic, how many particles do you “hit” along that line of sight. Through the edges, you can actually see “through” the star meaning that you see the photons from the edge and some of the ones behind that material penetrate through, but the empty space behind it doesn’t shine any photons through. The view through the center of the star is quite optically thick (hard for photons to get through all the material), but even some penetrate through, making the center actually brighter. Because of the high optical depth, this is only really noticeable around the limbs, I believe.
Forgot to also mention that limb darkening also results because the temperature near the surface of the star is lower than the center–so less photons come from the edge material than the stuff from the center. Because of the optical depth photons going from the center have to pass, this isn’t a huge overall effect, but it adds to the limb darkening. http://en.wikipedia.org/wiki/Limb_darkening actually explains it nicely.
I’m not sure this would apply straightforwardly to a gaseous body like the sun, but the light emission from most glowing surfaces is not equal in all directions. Most commonly it’s Lambertian, emitting more light per surface area in a direction normal to the surface than at other angles.
http://en.wikipedia.org/wiki/Lambert's_cosine_law
I circulated this question on Google+, and a helpful astronomer produced the answer… which goes by the name of “limb darkening”.
Essentially one can can “see through” the edge of the star, but not the centre.
http://en.wikipedia.org/wiki/Limb_darkening
I do not know whether this is relevant to your question, but it is my understanding that the apparent size of the image of any but the closest stars in any telescope is the result of imperfections in the optics and/or stability of the telescope’s platform and aiming apparatus, rather than a picture of an actual disk.
In addition, even the hypothetical, currently-unobservable disk of most stars under examination would almost always be smaller than the size of a single pixel in the light-detecting array.
Ralph
Thanks everyone for your comments. The reason I was puzzled was that, for the sun (unlike the moon), I couldn’t see a mechanism that would exactly balance out the opposing factors and give a uniform disc; but if it isn’t uniform, that problem doesn’t have to be faced.
A star (other than the Sun) is smaller than a single pixel, but the limb-darkening effect should still presumably be observable. When a planet transits the star, the intensity should decrease smoothly and then increase again rather than suddenly drop to a fixed value and then rise again. Do observations of extrasolar planets bear this out?
Not really. If a surface is more sloped away from your eye you see more square meters of it per solid angle, but less of the light is “aimed at you”, and these effects exactly cancel.
This is a general fact about geometry – and physics, but mainly geometry. If you have a solid surface emitting radiation at a constant temperature, it’ll have the same surface brightness at each point no matter how the surface wiggles. Here “surface brightness” is roughly the power per solid angle in the form of visible light. In simpler terms, it’s the brightness of a tiny pixel in a photograph of the object.
I’m having trouble finding a good online explanation, but this page tries to get the idea across:
http://www.cv.nrao.edu/course/astr534/Brightness.html
Limb darkening is an extra effect on top of this, coming from the fact that the Sun is not solid, but a gas, which glows more strongly further down. This makes the very edge of the Sun look somewhat dimmer.
Another cute fact is that no matter how far a star is, the observed surface brightness is the same:
http://mysite.verizon.net/vze55p46/id18.html
By the way, I hope everyone knows the difference between luminance, luminosity, luminous intensity, radiance, radiant intensity, irradiance, brightness and surface brightness. 🙂
It can get pretty confusing, but here’s a start:
http://en.wikipedia.org/wiki/Light_intensity