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Recently, I was working on the various astronomies and astrologies of my fantasy world, when I had an interesting thought: “What if there was a planet that was so faint, people would be arguing if it existed?”.

So far, I’ve just been befuddled. I’m certain if I dedicated myself, I’d figure it out for myself, but I would like to ask:

How do I determine when a planet is visible?

Is there some formula that I can use to know when a planet could be seen by the naked eye?

I know that a lot of things could affect when an astronomical body would be visible, including orbit eccentricity (of both the earth and the planet), the body’s size, its color, the brightness of the star, the thickness of the atmosphere, how close it is to the earth, and so on.

For context, I would like the faint planet (which orbits father out from the star) to be just barely visible to the naked eye at sea level under ideal conditions. As I understand it, that means that a) the planet could only be seen from sea level when it was closest to the earth, b) the atmosphere would impact visibility, so the planet could be seen more clearly and for longer at higher altitudes, and c), the atmosphere might impede visibility enough that the faint planet could only be seen at the zenith. (Also, human sight varies enough that what may be clearly visible to one could be completely invisible to another standing next to him. How would I account for that?)

So, to reiterate, what formula can I use to determine when and for how long such a planet is visible?

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    $\begingroup$ Hi and welcome to the forum! This is a great question, and you are right! There are a lot of factors. It’s a little more than just math, but I think you will get some good response in here. $\endgroup$
    – Vogon Poet
    Mar 1 at 3:39
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    $\begingroup$ In real life, such a planet is Uranus, whose apparent magnitude varies between 5.4 (visible in dark skies with good eyes) and 6 (mostly not visible). Another example is the (minor) planet Vesta; it is the brightest asteroid and it is sometimes, rarely, visible with the naked eye. $\endgroup$
    – AlexP
    Mar 1 at 3:49
  • $\begingroup$ @AlexP ...and Uranus was not known to classical astronomers. AFAIK nobody tried to catalog stars down to that magnitude until the 19th or 20th Century. And even if they did, the apparent motion of Uranus is so slow that it might not have been recognized as a planet. So then the question changes from how it could be seen to how it might be noticed. $\endgroup$ Mar 2 at 19:05
  • $\begingroup$ @TechInquisitor: That neither Uranus nor Vesta were known to the ancient astronoments although they are mostly (Uranus) or sometimes (Vesta) visible with the naked eye is kind of the entire point, isn't it? (And Ptolemy's star catalogue did include most of the stars visible with the naked eye from Alexandria, down to the 6th magnitude. Ptolemy even boasts that his catalogue includes "as many stars as it was possible to perceive, even to the sixth magnitude". Urban light pollution was not a big thing in the 2nd century CE.) $\endgroup$
    – AlexP
    Mar 2 at 20:19
  • $\begingroup$ @AlexP The point is that this fact entirely changes the question. It's no longer a matter of how such a thing can theoretically exist, because it actually exists. The question becomes how to explain it ever being noticed, if not how it was actually noticed in our world. $\endgroup$ Mar 2 at 20:34

4 Answers 4

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All the formulas assume that we are speaking about our real world solar system. In another solar system, you need to take into account the different luminosity of the star.

When speaking about the luminosity of a celestial body we use the word magnitude. Long time ago, when the ancient astronomers looked at the points of light dotting the night sky, they noticed that some were brighter than others, and they classified them into categories of brightness; the brightest ones were classified in the first magnitude, and so on down to the sixth magnitude, which grouped the faintest and barely visible stars.

Nowadays we use a much more strictly mathematical definition, but the basics remained the same: the smaller the magnitude, the brighter the object; object with magnitudes less than 5 are normally visible in the night sky by people with normal vision; objects with magnitudes between 5 and 6 are visible only by people with good vision and on dark, not light-polluted skies; and objects with magnitudes greater than 7 are not visible with the naked eye.

Magnitudes are logarithmic, so that a difference of 5 magnitudes corresponds to a change in brightness by a factor of 100; that is, a difference of 1 magnitude corresponds to a change in brighness by a factor of $\sqrt[5]{100} \approx 2.52$. The weird factor was chosen because modern astronomers wanted to preserve the ancient magnitude assignments to the star as much unchanged as humanly possible.

Note that as magnitude increases, brightness decreases. For example, the Sun has (apparent) magnitude −26.7 (that's minus twenty-six point seven); the full moon has (apparent) magnitude −12.74; Venus, the brightest planet, has (apparent) magnitude −4.2; Jupiter has on the average an (apparent) magnitude of −2.2; Sirius, the brightest star, has (apparent) magnitude −1.46; Mars has on the average an (apparent) magnitude of 0.7; the North Star, Polaris, has an apparent magnitude of 2.

  • Apparent magnitude less than 3: visible even from light-polluted urban landscapes.
  • Apparent magnitude between 3 and 5: easily visible on dark skies.
  • Apparent magnitude between 5 and 6: visible on dark skies for people with normal vision.
  • Apparent magnitide between 6 and 7: visible to some people on dark skies with no light pollution at all.
  • Apparent magnitude over 7: not visible with the naked eye, no matter how dark the night and how good the eyes.
  1. Determine the absolute magnitude of the planet.

    • The phrase "absolute magnitude" has two different meanings, depending on whether we speak of stars or of bodies in the solar system. Here I mean the absolute magnitude in the sense applicable to bodies in the solar system.

    • For an object in the solar system, the "absolute magnitude" $H$ is the apparent magnitude the object would have if it was at 1 A.U. from both the Earth and the Sun, and we saw it fully illumintated by the Sun. (Which is obviously impossible from geometric considerations.)

    • For a celestial body in the solar system with a diameter of $D$ kilometers and a geometric albedo $p$, assuming that the planet reflects light as a nice smooth diffuse spherical reflector, the absolute magnitude $H$ is given by

      $$H = 5 \log_{10} \frac {1329}{D\sqrt p}$$

    • Let's apply this to Jupiter; the diameter is about 140,000 km and the geometric albedo is about 0.54:

      $$H = 5 \log_{10} \frac {1329}{140{,}000 \sqrt {0.54}} \approx -10.8$$

      (The approximation of the planet as a nice smooth diffuse spherical reflector is not too bad for a gas giant, or a planet covered in clouds like Venus; but fails spectacularly for an airless or almost airless body such as the Moon or Mars, which are most certainly not smooth.)

  2. Determine the apparent magnitude of the planet as seen from Earth.

    • The apparent magnitude $m$ of a body in the solar system with absolute magnitude $H$, situated at a distance $d_O$ from the observer and a distance $d_S$ from the Sun, as seen by an observer situated at a distance $d$ from the Sun, is given by

      $$m = H + 5 \log_{10} \frac{d_S d_O}{d^2} - 2.5 \log_{10} q(\alpha)$$

      where $\alpha$ is the phase angle, that is, the angle between the body-to-the-sun and body-to-the-observer lines; and $q(\alpha)$, a number between 0 and 1, is the fraction of reflected light, depending obviously on the phase angle but also on the properties of the reflector. As above, if we assume that the planet is a nice smooth diffuse spherical reflector,

      $$q(\alpha) = \frac 23\left(\left(1-\frac\alpha{\pi}\right)\cos\alpha + \frac1\pi\sin\alpha\right)$$

      with $\alpha$ in radians. Note than in the best position, when the Earth is directly between the Sun and the planet, $\alpha$ is zero so that $q(\alpha)$ is 2/3.

    • Let's apply this to Jupiter (whose absolute magnitude we computed as about ˗10.8 above), when it is at the minimum distance from Earth; we have $d = 1$ (in A.U.), $d_S = 5$ (in A.U.), $d_O = 4$ (in A.U.), and in this position $\alpha = 0$ so $q(\alpha) = 2/3$.

      $$\begin{multline}m = H + 5 \log_{10} \frac{d_S d_O}{d^2} - 2.5 \log_{10} q(\alpha) \\ = -10.8 + 5 \log_{10} 20 - 2.5 \log_{10} 2/3 \approx -3.85\end{multline}$$

      For real, the smallest (that is, brightest) apparent magnitude of Jupiter is −2.94. Our brutal oversimplifications came within one magnitude of reality, which is good enough for a quick estimation.

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  • $\begingroup$ Can you please give a way to calculate the Geometric Albedo? Can't find any formulae or anything online. $\endgroup$
    – Zoey
    Mar 2 at 3:54
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    $\begingroup$ @Zoey: it is observational, not calculated. There is a nice table in the Wikipedia article has a nice table with several bodies in the solar system. Pick one which is similar. (Note that for nice smooth round gas giants it is about 0.5.) (You want the visual geometric albedo column.) $\endgroup$
    – AlexP
    Mar 2 at 9:48
  • $\begingroup$ I saw that page, it doesn't explain what the " Albedo of an idealized flat, fully reflecting, diffusively scattering (Lambertian) disk with the same cross-section" is. I am just wondering how you find that. $\endgroup$
    – Zoey
    Mar 2 at 21:02
  • $\begingroup$ A Lambertian reflector is simply a perfect diffuse reflector; that is, it reflects all the incoming light isotropically, so that the observed intensity of the light does not depend on the angle of view of the observer but only on the angle between the direction incident light and the normal to the surface. All it does is introduce the factor of 2/3 in the theoretical calculation of $q(\alpha)$ (because integrating over a sphere). $\endgroup$
    – AlexP
    Mar 2 at 21:18
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A good start is to notice which planets were known to humans before telescopes. I hope this works. It should show you highlighted text showing that the Sun, Moon, and Mercury through Saturn were known before telescopes.

The visibility of a planet depends on how far away it is, how big it is, and what the outer surface or top of the atmosphere is made of. Mercury is close to the Sun, so at the right times it is quite visible. Venus has a lot of white clouds, so is easily visible. Saturn has its rings so is easily visible even though it is much farther away. Here are the albedos for the planets. Note that the moon is actually not all that reflective, with an albedo of 0.12. If it was not so close it would be hard to see. Neptune is farther away and less reflective than Saturn, so requires a telescope to see, even though it is one of the larger planets.

You could vary that a little bit by having your protagonist have either better than or worse than 20/20 vision. There was a famous vision test from ancient times. The second star from the end of the handle of the Big Dipper is a double star. Ancient people used ability to see this star as a test of good vision. So, you could have it so only people with very good vision are able to see your farthest planet. And maybe they get some special status as “sharp eyed.” Maybe that planet gets a name meaning “very good vision.” That could be a very nice little “business” in a story. Maybe other planets are named for other characteristics. Along the lines of Mercury for change, Venus for love, Mars for war, Jupiter for being jovial.

It must be the right point in the orbit of the home planet of your protagonist, and the right spot for the other planet. If the two planets are on opposite sides of the Sun, then they won’t be able to see anything. And if the orbits are very eccentric then they won’t be able to see the planet when it is on the outer leg of its orbit (the technical word for that point is the apastron). With Saturn having a 29 year orbit, if it had an apastron somewhat farther it might only be visible for, say 10 years, then invisible for 20 or so. And, of course, you would need to consider the moon. If your world includes a moon. When the moon is out it is tough to see even the brightest stars through the light it reflects.

Planets closer to the Sun are only visible at dusk or dawn. If you think about the geometry, they cannot be far from the sun or and are not visible in bright daylight. So, you get Venus as “the morning star” and the “evening star.” Some ancient people thought these were different planets.

It is odd you specify "at sea level." Visibility will be better at altitude. That might be another nice little "business" in a story. The mountain people and the valley people might have sharply different opinions about whether a given planet exists. There could be a lot of drama there with the valley people believing the mountain people were mad or something.

At this link you can find a table with the orbit radius and length of year for each planet. You can see some fairly regular patterns. And you can see why there is a "missing" planet between Mars and Jupiter. If you just "winged it" with this table, you could do OK. So if you want something with an orbit about 12 years, put it at Jupiter's orbit. If you want 20 years, you put it about half way between Jupter and Saturn. And so on.

Last thing. If you want to play with eccentric orbits, you can use the "equal area" law to get the time it spends on each part of the orbit. You join the planet to the sun with an imaginary line and think about the area this line sweeps out during the orbit. It makes equal area wedges during each month. (Or whatever length of time you break up its year.) When the planet is farther away from the sun the wedge is long and thin. When it’s close to the sun, it moves faster, making the wedge short and wide.

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This kind of discussion goes nowhere. Since different people have different levels of good or bad vision, those with poor eyesight will just take the word of people with good eyesight at face value. It is similar to that sound frequency that old people cannot hear.

People are intelligent, and once they figure out the laws of gravitation they will be tracking planets, trying to guess where those celestial bodies will be next. It is not as simple as it seems, there is a lot of math involved. And then they see that some planets don't follow the paths they're supposed to be on, so something must be pulling them off their tracks.

This is how Neptune and Pluto were discovered, too long before they could even be seen, due to their influence over other bodies. This sparks some discussion whether a planet is there or not, as well as a race to get a picture and naming rights.

And then there is Vulcan, a planet that was expected to exist in an orbit lower than Mercury. Mercury's orbit was considered too messed up and the only explanation people people had for a long while was another planet accelerating it from below. There was a race to find that planet, which got a temporary name until a picture could be taken. It was Einstein who figured that Mercury was movong so fast relative to us that relativity kicked in, and accounting for that finally allowed us to keep tabs on Mercury properly. So if your world knows Kepler's and Newton's laws but not relativity, stargazers might be on a wild goose chase for a body that is not there.

It is quite feasible that a guild of mages or priests from some church might be having these discussions among them. The populace would knpw no better, and would mention the missing planet just like real world 21st century people do with quantum science - just to make conversation. And since people in our own world keep hallucinating phenomena in the sky all the time, it is feasible that in your world people will "see" the missing planet either out of fervor or intoxication.

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  • $\begingroup$ I think finding Pluto was a lucky fluke. It is nowhere near massive enough to account for the discrepancy in orbits that triggered the search for it. I also vaguely recall something about that discrepancy being an error in the first place. $\endgroup$ Mar 2 at 19:19
  • $\begingroup$ @MichaelRichardson you are 100% right there, but the wrong math led to the discussion of something there and the eventual discovery of Pluto anyway. $\endgroup$ Mar 2 at 20:27
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I see several questions, I'll attempt to answer the one without the formulas,

Q: “I was working on the various astronomies and astrologies of my fantasy world, when I had an interesting thought: “What if there was a planet that was so faint, people would be arguing if it existed?”.

Patience, a sharp eye and no cultural controversies among astrologers

It requires patience to wait for movement

Among astronomers and astrologers, there has always been only one criteria to call something a planet: this faint spot in the sky must be moving. If you can provide proof it is actually moving, you had your planet discovery..

It depends on the patience and the distance of the new planet to the sun. For a discovery like Saturn (movement: 1/10th of a degree per month, see ephemeris) you'll need a few years to observe it and actually see it move!

https://in-the-sky.org/ephemeris.php?objtxt=P6

But suppose you have 2-3 people with a sharp eye, who are able to look up and have patience, when they know the fixed stars by constellation, a moving spot will be noticed by multiple people. Also, if it very faint.

At some point, astrologers will agree and try to predict position and assign meaning

In a world without telescopes, if you want to discover a planet and agree upon its existence, you'll just need a few savants with knowledge of a calendar and the night sky constellations. And a sharp eye. When you have e.g. 2 astrologers with a sharp eye, who both confirm that a new planet exists, their community or temple of priests/astrologers will agree upon its existence.. and start a debate about the meaning of the new planet.

The existence could become a controversy amongst astrologers

Astronomers would provide the truth of course.. they can decide objectively.

But suppose your astrologers would set the agenda. Different astrologers could have assigned different meanings to this new planet. In that case, the interpretation (meaning) assigned to the planet could raise cultural controversy, where some simply don't believe the planet exists, because its meaning represents a taboo, or some other difficulty. Solely on cultural grounds they won't accept the faint dot as a planet! Amongst astrologers, you can have that kind of misunderstandings.

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