<p>The other answers mostly make unstated assumptions, which are not given in the original questions. Such as implying, &quot;as visible to the naked eye&quot; or &quot;ignoring situations where the sun's glare makes it hard to see.&quot;</p>
<p>Let's approch the question only considering how much light from the plants reach earth, not considering how easy it is to observe.</p>
<p>Background: As mentioned in some comments:
During transits, both mercury and venus will be invisible except as dark spots blocking a bit of the sun.  When at their closest approach to the sun during their years when no transit occurs, they will be completely invisible from earth because of the sun's glare and they will have very tiny crescent fractions illuminated as seen from earth.  These two planet may well be visible from some of the solar observatory satellites, such as SOHO. I just checked and SOHO can indeed see mercury and venus when very close to the sun, such as <a href="https://soho.nascom.nasa.gov/hotshots/2000_05_03/" rel="nofollow noreferrer">https://soho.nascom.nasa.gov/hotshots/2000_05_03/</a> but in that photo the two planets are nearly full as seen from near earth.  Therefore, both venus and mercury's illuminated areas will range from an arbitrarily small fraction, when approaching the sun to zero when transiting to very nearly 100% illuminated when opposite the sun.</p>
<p><strong>Maximum</strong>:
At the brightest, mercury and venus are both visually quite bright.
At maximum brightness, venus is -4.92 and mercury is -2.48 magnitude (<a href="https://en.wikipedia.org/wiki/Apparent_magnitude" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Apparent_magnitude</a>).  This means venus is 9.5 times brighter, as seen from earth. (using the flux ratio formula from <a href="http://burro.case.edu/Academics/Astr221/Light/magscale.html" rel="nofollow noreferrer">http://burro.case.edu/Academics/Astr221/Light/magscale.html</a>)</p>
<p><strong>Minimum</strong>:
When at minimum brightness the light coming from the planet won't actually be visible, but will represent how much light reflected from the planet gets to us when the planet is 0% illuminated by the sun - that is during transit. This brightness will be determined by:</p>
<ol>
<li>how much starlight and light from non-sun solar system objects, like comets, dust, planets, illuminates the planet.</li>
<li>the distance from the planet to earth (at transit), which will be the minimum earth-venus or earth-mercury distance</li>
<li>the albedo</li>
<li>the size of the planet</li>
</ol>
<p>Factor 1 will be about the same for both.</p>
<p>Factor 2: For dimmest light at earth, consider the greatest earth-planet distance during transit. Let's pretend transits can sometimes occur when the planet is at it's closest to sun (perihelion) and earth is at farthest (aphelion).  For mercury and venus, perihlion distances are 46,001,200 km and 107,477,000 km, respectively.  The greatest earth-sun aphelion is 152,097,597.  So the relevant distances to earth are 106,096,397 for mercury and 44,620,597 for venus.</p>
<p>Factor 3: The albedos are 0.12 and 0.75 for mercury and venus, respectively (<a href="https://astronomy.swin.edu.au/cosmos/a/Albedo" rel="nofollow noreferrer">https://astronomy.swin.edu.au/cosmos/a/Albedo</a>)</p>
<p>Factor 4: the radius of mercury is 2,439.7 and that of venus is 6,051.8 km.</p>
<p>To combine the factors above, the ratio of the minimum brightness of mercury to the minumum brightness of venus will be:<a href="https://i.sstatic.net/CxchP.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/CxchP.png" alt="ratio of the minimum brightness of mercury to the minumum brightness of venus" /></a></p>
<p>with numbers</p>
<p><a href="https://i.sstatic.net/L1kuw.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/L1kuw.png" alt="ratio of the minimum brightness of mercury to the minumum brightness of venus with numbers" /></a></p>
<p>So at their brightests, venus is 9.5 times brighter than mercury.
At their faintests, venus is 1/0.00224 = 447 times brighter than mercury.  So the brightness of venus changes by a greater <em>factor</em> than that of mercury.</p>
<p>That is a <em>relative</em> difference, if want an <em>absolute</em> difference, venus also changes by more.  From near zero light to -4.92 magnitude for venus, compared to near zero light to -2.48 magnitude.</p>
<p>So the answer is <em>venus</em> for either a difference or a factor (ratio) of change between maximum and minimum brightness.</p>

CC BY-SA 4.0
2023-05-05T18:00:46.303Z
2023-05-05T18:00:46.303Z
William Boyer
<p>What is the planet relative to earth that shows the greatest change in apparent brightness?</p>

<p>I think the answer is Mars, and a quick google search reveals it is Mars.</p>

<p>I have my reasoning here, but I am not sure if I am correct.</p>

<p>One way to do this problem is to get all the planets and collect their perihelion and aphelion distances from the sun, and see the different.</p>

<p>However I want to be able to explain this without the extensive use of numbers. </p>

<hr>

<p>Here is my explanation:</p>

<p>The closest planet to us is Mars, both in terms of its aphelion and its perihelion distances</p>

<p>The Earth-Mars distance varies between 2.66 AU and .38 AU</p>

<p>Mercury is almost always blinded by the sun, so we rarely see it in the sky</p>

<p>Venus is frequently viewed as a crescent, and thus its apparent brightness also dimishes</p>

<p>Jupiter and rest of the superior planets are simply too far away for us to notice a big differene</p>

<hr>

<p>Are all my points correct here?</p>

CC BY-SA 4.0
2018-10-07T02:15:05.043Z
What is the magnitude of Mercury or Venus during a transit? Those may be the faintest planet and therefore the winner :-)
JohnHoltz
<the-sun><planet><apparent-magnitude>
What is the planet relative to earth that shows the greatest change in apparent brightness?
K Split X
# What is the planet relative to earth that shows the greatest change in apparent brightness?

(Asked by: username_0 on 2018-10-06)

What is the planet relative to earth that shows the greatest change in apparent brightness?

I think the answer is Mars, and a quick google search reveals it is Mars.

I have my reasoning here, but I am not sure if I am correct.

One way to do this problem is to get all the planets and collect their perihelion and aphelion distances from the sun, and see the different.

However I want to be able to explain this without the extensive use of numbers.

---

Here is my explanation:

The closest planet to us is Mars, both in terms of its aphelion and its perihelion distances

The Earth-Mars distance varies between 2.66 AU and .38 AU

Mercury is almost always blinded by the sun, so we rarely see it in the sky

Venus is frequently viewed as a crescent, and thus its apparent brightness also dimishes

Jupiter and rest of the superior planets are simply too far away for us to notice a big differene

---

Are all my points correct here?

**Tags:** the-sun, planet, apparent-magnitude
> **Comments:**
> **username_1**: What is the magnitude of Mercury or Venus during a transit? Those may be the faintest planet and therefore the winner :-)  

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## 1 answers

### Answer by username_2 on 2023-05-05. Score: 2

The other answers mostly make unstated assumptions, which are not given in the original questions. Such as implying, "as visible to the naked eye" or "ignoring situations where the sun's glare makes it hard to see."

Let's approch the question only considering how much light from the plants reach earth, not considering how easy it is to observe.

Background: As mentioned in some comments:
During transits, both mercury and venus will be invisible except as dark spots blocking a bit of the sun. When at their closest approach to the sun during their years when no transit occurs, they will be completely invisible from earth because of the sun's glare and they will have very tiny crescent fractions illuminated as seen from earth. These two planet may well be visible from some of the solar observatory satellites, such as SOHO. I just checked and SOHO can indeed see mercury and venus when very close to the sun, such as <https://soho.nascom.nasa.gov/hotshots/2000_05_03/> but in that photo the two planets are nearly full as seen from near earth. Therefore, both venus and mercury's illuminated areas will range from an arbitrarily small fraction, when approaching the sun to zero when transiting to very nearly 100% illuminated when opposite the sun.

**Maximum**:
At the brightest, mercury and venus are both visually quite bright.
At maximum brightness, venus is -4.92 and mercury is -2.48 magnitude (<https://en.wikipedia.org/wiki/Apparent_magnitude>). This means venus is 9.5 times brighter, as seen from earth. (using the flux ratio formula from <http://burro.case.edu/Academics/Astr221/Light/magscale.html>)

**Minimum**:
When at minimum brightness the light coming from the planet won't actually be visible, but will represent how much light reflected from the planet gets to us when the planet is 0% illuminated by the sun - that is during transit. This brightness will be determined by:

1. how much starlight and light from non-sun solar system objects, like comets, dust, planets, illuminates the planet.
2. the distance from the planet to earth (at transit), which will be the minimum earth-venus or earth-mercury distance
3. the albedo
4. the size of the planet

Factor 1 will be about the same for both.

Factor 2: For dimmest light at earth, consider the greatest earth-planet distance during transit. Let's pretend transits can sometimes occur when the planet is at it's closest to sun (perihelion) and earth is at farthest (aphelion). For mercury and venus, perihlion distances are 46,001,200 km and 107,477,000 km, respectively. The greatest earth-sun aphelion is 152,097,597. So the relevant distances to earth are 106,096,397 for mercury and 44,620,597 for venus.

Factor 3: The albedos are 0.12 and 0.75 for mercury and venus, respectively (<https://astronomy.swin.edu.au/cosmos/a/Albedo>)

Factor 4: the radius of mercury is 2,439.7 and that of venus is 6,051.8 km.

To combine the factors above, the ratio of the minimum brightness of mercury to the minumum brightness of venus will be:[![ratio of the minimum brightness of mercury to the minumum brightness of venus](https://i.sstatic.net/CxchP.png)](https://i.sstatic.net/CxchP.png)

with numbers

[![ratio of the minimum brightness of mercury to the minumum brightness of venus with numbers](https://i.sstatic.net/L1kuw.png)](https://i.sstatic.net/L1kuw.png)

So at their brightests, venus is 9.5 times brighter than mercury.
At their faintests, venus is 1/0.00224 = 447 times brighter than mercury. So the brightness of venus changes by a greater *factor* than that of mercury.

That is a *relative* difference, if want an *absolute* difference, venus also changes by more. From near zero light to -4.92 magnitude for venus, compared to near zero light to -2.48 magnitude.

So the answer is *venus* for either a difference or a factor (ratio) of change between maximum and minimum brightness.


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