## New Problem: Surface Brightness

When I began studying gravitational lensing, I was told that a gravitational lens preserves a quantity called “surface brightness”. This is defined as the flux per unit area. In other words, if you look at the sun from out at Pluto, it’s very dim. But if you receive $\frac{1}{1000}$ as much light at Pluto, it’s because the size of the sun in the sky is $\frac{1}{1000}$ what it is here, not because it’s gotten intrinsically dimmer. The stars are just as bright here as they are right up close. They look really small, though (so small even a big space telescope like Hubble can’t see any details on a star (except the Sun, of course)).

Even though gravitational lensing can bend light and thereby make a star seem bigger in the sky, it cannot make it seem intrinsically brighter. The same is true for normal optics. A magnifying glass can make the words on a page larger, but if the lights in the room are dim the magnifying glass cannot make anything brighter.

Question: why not?

Hint: you don’t need to know anything about optics or gravity to answer this question, except that lenses and gravitational potential are completely passive. That is, they only bend light, not create it or change it.

### 7 Responses to “New Problem: Surface Brightness”

1. Nik Says:

To increase the brightness of an image, you should increase the amount of light being received from the object. So, you’d have to increase the inherent brightness (luminosity) of the star or move closer to it (apparent brightness is inversely proportional to the square of the distance to the light source). Since gravitational lensing doesn’t do either, the intrinsic brightness of the star can’t change.

Technically, I just repeated everything you said in the question.

2. meichenl Says:

You actually CAN increase the light received – that’s magnification. Total light received from an object can be thought of as the result of a multiplication.

light = brightness * size

“brightness” is the light per unit steradian, and “size” is the steradians subtended by the object. (steradians are essentially “two-dimensional angles”. just as 2*pi radians is a full circle, 4*pi steradians is the entire sky. the 4*pi comes from the area of a sphere being 4*pi*r^2)

so light could increase because of an increase in brightness, and increase in size, or both. the mystery is, why does it only increase in size in real life?

3. meichenl Says:

also, sorry it takes me so long to get back to your comments. wordpress seems to notify me of comments only when it feels so inclined, or else yahoo thinks emails about your comments are spam.

4. Answer: Surface Brightness « Arcsecond Says:

[…] Answer: Surface Brightness As always, check out the question. […]

5. Nik Says:

This defies any common sense. How can magnification increase the amount of light received? That’s like saying the number of photons magically multiply after/during magnification.
Maybe I miss understood what you said, because what I gathered is that the amount of light per unit steradian is invariable. Therefore, if the total steradians subtended increases so does the light. How?!! Where did the “extra light” come from?
So, the first thing I thought was that light received would be the constant and size and brightness would be inversely proportional.

A lens (convex) only concentrates light over a particular area, as you said it doesn’t create it.
Consider a fairly large convex lens (perfect lens) in a pitch dark room. The object is a candle at a point beyond the center of curvature of the lens. Now, the image (real) will be magnified ie greater steradian angle subtended. How could the brightness of the image flame equal that of the object flame?

I’m missing something very obvious here. :/

6. Nik Says:

Ah, spam. Heartwarming.

7. meichenl Says:

You’re increasing the light received by a single observer, at a specific location. Somewhere else, the total light received would have to go down to compensate. As you say, the total number of photons remains constant.