Light intensity from a distant star obeys the distance-squared law. What if the star were a filament instead?

Question

If our distance from a light-radiating body, that (like a distant star) is nearly a point source, is doubled, the intensity of light received from it becomes (nearly exactly) one quarter of what it was before.
Suppose that it was possible for a star to be stretched across space in a thin incandescent filament that, for all intents and purposes, was endless.
What kind of mathematical law would then relate the intensity of light reaching us from this filament to our distance from it?

Answer 1

The light intensity would fall off as 1/r instead of 1/r^2, where r is distance. I assume the the filament length L is very long compared to the distance of the observer r. This follows from conservation of radiation energy. The area of cylinder of fixed length L and variable radius r with this condition (L>>r) increases in proportion to r, while the area of a sphere of radius r increases in proportion to r^2. Intensity at radius r is radiated power divided by this area for a cylindrical ("filament") and spherical source, respectively.

I see we have some responders here who memorize formulas without understanding their context. You see this a lot with inverse-square law problems. It's probably because such laws are introduced so early in physics courses.

Answer 2

Treat the inverse square law as for point sources, then plot you line with x,y coords. Take a small section of the line dx say the inverse square law applies. Then integrate along the length of the line between the start and end points of the star. This integral will give you the law for a filament of that length, you can integrate between minus and plus infinity if you like and I'd expect if you do the integration correctly that the relationship will reduce to the point like when you take the length to be zero.

As far as I can make out (and it's been a long time since I did calculus) it's then the sum of the integral of k(y^2+x^2)dx over the region x=a to x=b where a and b are the start and end points on the x axis and y is the perpendicular distance to the filament and k is the proportionality constant.

I get
Integral=(1/y)arctan(x/y)
so I=(1/b)arctan(y/b)-(1/a)arctan(y/a) but it's been so long and I remember doing this sort of thing for gravity of a filament of mass and that doesn't look right.

The answer above saying 1/r would be correct if it was a 1D universe and a point but the point closes to you will give this contribution, the point next to it increases the overall intensity.

Answer 3

The nearest star is 4.2 light years away, not 50,000. But you are right, things can happen to light as it travels through space. Most commonly, some of it is absorbed by gas and dust between here and there. But space is mostly empty, so unless there is a nebula in the way, this effect is very small in most cases. Another thing that happens to light is that light from very distant galaxies is shifted toward lower frequencies by the expansion of space. Any motion of the star towards or away from us will cause a shift in wavelengths, though for stars in our own galaxy this is much smaller than the red-shift of distant galaxies. Magnetism and gravity can also cause slight wavelength shifts. The way we can tell what happens to the star's light is by comparing it to other stars. All stars have distinctive dark lines in their spectra at known wavelengths, so we can compare what the wavelength is to what we can calculate it should be. We can also identify the true brightness of some stars, and compare that against observed brightness to see if any light has been absorbed. It's impossible to move at or beyond the speed of light, so you're right, we would have to do something else if we hope to reach other stars. Supposedly wormholes can connect two distant points in space, but no one has been able to figure a way they could be of any practical use. The physics of that, if it's possible, is beyond anything we currently know.

Answer 4

The inverse squared law will still apply. But if you stretched the star into a thin filament in absolute terms (not from where we see it), it's light will diminish, and depending on what you mean by 'a filament', it could be so thin that there will be only very faint light, if anything at all. If you stretched any star till it was an atom thick, it would not be able to emit any light long before it got so thin anyway.

Answer 5

Exactly the same . Light is light, irregardless of its source.