<p>The brightness of reflected light of the planet depends on the brightness of the incident light and the distance (d) from the planet to the observer.</p>
<p>Light obeys an inverse square law (think of light as spreading out in a sphere) So the brightness of the planet is inversely proportional to the distance from the planet to the observer (B is proportional to <span class="math-container">$\frac{1}{d^2}\ $</span>)  (assuming things like the planet is fully illuminated etc)</p>
<p>But the brightness of the incident light also obeys an inverse square law with respect to the distance from the planet to the sun (r). The brightness of the incident light is proportional to <span class="math-container">$\frac{1}{r^2}\ $</span></p>
<p>So the brightness is proportional to <span class="math-container">$\frac{1}{d^2}\ $</span>×<span class="math-container">$\frac{1}{r^2}\ $</span>
But if your observer is near the sun, then d=r and so the brightness is proportional to <span class="math-container">$\frac{1}{r^4}\ $</span></p>

CC BY-SA 4.0
2022-07-27T18:35:03.320Z
To paraphrase: as a planet moves, $ \frac{1}{r^2}\ $ light from the Sun reaches the planet.  Then $ \frac{1}{r^2}\ $ is of the reflected light reaches back to the Sun.
Greg Miller
CC BY-SA 4.0
2022-07-28T07:39:35.870Z
One doubt: Can't we just apply 1/(2r)^2
Particle king
CC BY-SA 4.0
2022-07-28T08:05:07.367Z
no.  That's not how fractions work
James K
CC BY-SA 4.0
2022-07-28T09:43:10.263Z
@Particleking 1/(2r)^2 would be correct for a big (at least half sun sized) flat mirror at distance r. But (unlike a mirror) a planet disperses the light in all directions, so it has to be treated as a new light source.
Ralf Kleberhoff
CC BY-SA 4.0
2022-07-27T15:57:08.230Z
2022-07-29T20:09:01.240Z
2022-07-29T20:09:01.240Z
James K
<blockquote>
<p>When viewed from the sun, the brightness of the planet with a given
size and albedo changes according to the fourth power of inverse
distance.</p>
</blockquote>
<p>I found the statement in one encyclopedia but I can't find any mathematical proof of this.</p>

CC BY-SA 4.0
2022-07-27T15:42:42.630Z
Look for "inverse square law"
Greg Miller
CC BY-SA 4.0
2022-08-01T16:45:29.473Z
Physics doesn't prove. It measures ;)
planetmaker
<luminosity>
Change in Luminosity of a planet
Particle king
# Change in Luminosity of a planet

(Asked by: username_0 on 2022-07-27)

> When viewed from the sun, the brightness of the planet with a given
> size and albedo changes according to the fourth power of inverse
> distance.

I found the statement in one encyclopedia but I can't find any mathematical proof of this.

**Tags:** luminosity
> **Comments:**
> **username_1**: Look for "inverse square law"  
> **username_2**: Physics doesn't prove. It measures ;)  

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## 1 answers

### Answer by username_3 on 2022-07-27. Score: 5 | Accepted answer

The brightness of reflected light of the planet depends on the brightness of the incident light and the distance (d) from the planet to the observer.

Light obeys an inverse square law (think of light as spreading out in a sphere) So the brightness of the planet is inversely proportional to the distance from the planet to the observer (B is proportional to $\frac{1}{d^2}\ $) (assuming things like the planet is fully illuminated etc)

But the brightness of the incident light also obeys an inverse square law with respect to the distance from the planet to the sun (r). The brightness of the incident light is proportional to $\frac{1}{r^2}\ $

So the brightness is proportional to $\frac{1}{d^2}\ $×$\frac{1}{r^2}\ $
But if your observer is near the sun, then d=r and so the brightness is proportional to $\frac{1}{r^4}\ $

> **Comments:**
> **username_1**: To paraphrase: as a planet moves, $ \frac{1}{r^2}\ $ light from the Sun reaches the planet.  Then $ \frac{1}{r^2}\ $ is of the reflected light reaches back to the Sun.  
> **username_0**: One doubt: Can't we just apply 1/(2r)^2  
> **username_3**: no.  That's not how fractions work  
> **username_4**: @Particleking 1/(2r)^2 would be correct for a big (at least half sun sized) flat mirror at distance r. But (unlike a mirror) a planet disperses the light in all directions, so it has to be treated as a new light source.  

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