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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?

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    $\begingroup$ What is the magnitude of Mercury or Venus during a transit? Those may be the faintest planet and therefore the winner :-) $\endgroup$
    – JohnHoltz
    Oct 7, 2018 at 2:15
  • $\begingroup$ Agree w/ @JohnHoltz, but, even when non-transiting, Venus and Mercury have phases and are extremely dim at their "new planet" phase. You may need to add some restrictions to "greatest change in apparent brightness" to avoid cases like this. As stated, I think Venus has the greatest change in brightness, partly because it goes from -4 magnitude to pretty much invisible, and Mars never gets anywhere near that bright. $\endgroup$
    – user21
    Oct 7, 2018 at 18:10

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$ \begin{array}{cccc} \text{Body} & \text{Max} & \text{Min} & \text{Delta} \\ \text{Sun} & -26.78 & -26.71 & 0.07 \\ \text{Mercury} & -2.45 & 5.58 & 8.03 \\ \text{Venus} & -4.89 & -3.82 & 1.07 \\ \text{Moon} & -12.87 & -3.76 & 9.11 \\ \text{Mars} & -2.88 & 1.84 & 4.72 \\ \text{Jupiter} & -2.94 & -1.66 & 1.28 \\ \text{Saturn} & 0.42 & 1.47 & 1.05 \\ \text{Uranus} & 5.31 & 5.95 & 0.64 \\ \text{Neptune} & 7.8 & 8. & 0.2 \\ \text{Pluto} & 13.75 & 15.96 & 2.21 \\ \text{Comet Halley} & 2 & 25.66 & 23.66 \\ \text{Tesla Roadster} & 6.66 & 29.29 & 22.63 \\ \end{array} $

The answer is Mercury (as above), subject to the proecedure/caveats below:

Procedure

  • I used HORIZONS to generate daily brightness data for a century for:

    • all the planets as viewed from Earth (except Earth itself)

    • the Sun, the Moon, Pluto, Comet Halley, and the Tesla Roadster

  • I then noted the minimum and maximum brigtness, along with the magnitude difference of these brightnesses, in the table above.

  • You can use HORIZONS to compute the results yourself, or view the results in the *-brightness.txt.bz2 files in https://github.com/barrycarter/bcapps/blob/master/ASTRO/

Caveats

  • Although Mercury's brightness changes more than Mars, the Sun's glare makes it impossible to see Mercury though Earth's atmosphere when Mercury's angular distance from the Sun is small. Therefore, Mars may be a better practical answer.

  • Because I used daily brightnesses, it's theoretically possible I missed absolute (intraday) minimums or maximums, especially for the Moon, whose brightness changes rapidly. However:

    • I took 100 years worth of data, and the moon's brightness doesn't have a period that's a multiple of one day. Since the moon's synodic period is approximately 29.5 days, its brightness is almost periodic in 59 days (2 synodic periods), but it's far enough from 29.5 days that this shouldn't be too much of a problem.

    • The Moon isn't a planet: I just added it for reference

    • Unless the Moon's actual brightness difference was higher than 22.64 magnitudes (which is probably unlikely[?]), it would remain in 3rd place in terms of brightness change, so the exact value isn't as important.

  • Because I used only a 100 year period, I did not include a complete orbit for either Neptune or Pluto. This shouldn't be an issue because:

    • The synodic period of both planets is just over a year, and much of the brightness change comes from Earth's own orbit, not Neptune's or Pluto's.

    • Even if the maximum magnitude change were slightly higher than in the table, it wouldn't make much of a difference.

  • Note that "Max" and "Min" refer to brightness, which is ordered the opposite of magnitude: lower magnitude means greater brightness.

  • Because I used daily data, I missed rare events such as transits and eclipses. The table above is for an "average" orbit, excluding special cases.

  • In some cases, HORIZONS gives "n.a." for magnitude data. I ignore these "n.a." values.

  • Data for the Tesla Roadster is only available from 2018-Feb-07 03:00 UTC to 2090-Jan-01 23:00 UTC, not the entire century.

Complexity

The problem is nontrivial. As you correctly note, the planet's geocentric and heliocentric distance play into the formula, but there's more to it.

Quoting Oliver Montenbruck and Thomas Pfleger's "Astronomy on the Personal Computer" (https://books.google.com/books?id=nHUqBAAAQBAJ):

enter image description here enter image description here

Paul Schlyter's http://www.stjarnhimlen.se/comp/ppcomp.html#15 provides similar nontrivial formulas.

EDIT: https://en.wikipedia.org/wiki/Phase_curve_(astronomy) provides more information on how planetary brightness varies nontrivially with planetary phase.

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  • $\begingroup$ Bravo; above and beyond! $\endgroup$
    – uhoh
    Oct 10, 2018 at 12:25
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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

with numbers

ratio of the minimum brightness of mercury to the minumum brightness of venus with numbers

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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