A few questions here have broached the possibility and survivability of life on a Trojan planet, a planet that exists at the Lagrange point of a larger celestial body. In this particular question, the habitability of the Trojan planet is taken for granted, and intelligent lifeforms on its surface are looking up toward the skies. What would they see?

  • The system contains a central star with 1.05 solar masses.
  • A brown dwarf of 0.043 solar masses/45 Jupiter masses is orbiting at 1.175AU.
  • An Earth analog with slightly less mass than our own planet is located at the brown dwarf's L5 Lagrange point.

The distance from the central star should make it appear nearly the same as our Sun does to us.

Question: How would the brown dwarf appear to the inhabitants of the Earth-analog planet?

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    $\begingroup$ (a) The question in paragraph #2 has nothing at all to do with Trojan planets, making the question title and the first paragraph distracting. (b) If the reference planet and the brown dwarf are in the same orbit ("... are around 1.175 AU...," which is remarkably specific for "around"). I've gotta assume they're orbiting at the same speed or one would crash into the other (or gravity would eventually bring them together), which means the brown dwarf is only visible (if it's visible) near twilight, right? If that didn't make sense, you need to clarify your question. $\endgroup$
    – JBH
    Commented Apr 27, 2023 at 6:18
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    $\begingroup$ A Trojan planet would share the same orbit as a large body, in this case a brown dwarf. The smaller planet is in a stable Lagrange point position, it would not crash into the other body. I don't understand how the 2nd paragraph has nothing to do with Trojans, I'm detailing the positions and masses of the bodies involved. $\endgroup$
    – Mahahus
    Commented Apr 27, 2023 at 6:25
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    $\begingroup$ Your question can be rephrased to "what would a brown dwarf look like as seen from a distance x, where x is the distance at which the observer is at the Lagrangian point of the brown dwarf-central star system?" $\endgroup$
    – L.Dutch
    Commented Apr 27, 2023 at 6:28
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    $\begingroup$ @Mahahus It had nothing to do with trojans because you did not (and, to a degree, still have not) clearly stated the relationships between these celestial bodies. You refer to an "earth analog" and a "rocky planet." At this time, I'm assuming they're the same thing. In fact (and I hope you'll forgive me), I'm going to edit your post to make it clear in terms of your current question and your comment. If you disagree with my edit, you can roll the changes back to the version of the question before my edits (you won't hurt my feelings). $\endgroup$
    – JBH
    Commented Apr 27, 2023 at 16:15
  • $\begingroup$ No by all means if you think it helps the question be more concise, I thought the relations were easily presumed but if it’s easier to read and get answers it doesn’t matter. $\endgroup$
    – Mahahus
    Commented Apr 27, 2023 at 20:44

2 Answers 2


The apparent brightness of a planet is related to its size and distance from its parent star. You've stated your brown dwarf has a mass of ~45x that of Jupiter, but planets much heavier than Jupiter are likely to just be a lot more dense and not actually that much larger. Wikipedia has a handy list of known brown dwarf stars including size and mass estimates, and you'll see that those around 40Mjupiter migh be as small as 0.8 jupiter-radii, or perhaps larger than 2 jupiter-radii, but mostly these things seem to be about the same size as Jupiter, which is handy.

The other figure you need to know but haven't stated is the brown dwarf's geometric albedo. I won't go into detail here, but I'll use Jupiter's value of ~0.538 for now.

Now, given that the brown dwarf is probably the same sort of size as Jupiter, and I've declared it to be the same sort of colour and reflectiveness as Jupiter, you can use the real Jupiter as a proxy for all further calculations. I'll also be using the normal Sun as a proxy for your star to make things easy.

The formula for apparent magnitude (how bright things look to an observer) is

$$m = H + 5\log_{10}\left({d_{BS}d_{BO} \over d_0^2}\right) - 2.5\log_{10}\left(q(\alpha)\right)$$

where $H$ is the brown dwarf's absolute magnitude, $d_{BS}$ is the distance between the primary star and the brown dwarf, $d_{BO}$ is the distance between the observer. In this case, they're both conveniently 1.175 AU. $d_0$ is a unit conversion factor, which in this case will be 1 AU because that's the distance unit we're using. The phase angle $\alpha$ between the planet and the brown dwarf will be a constant 60° thanks to their trojan relationship.

Because planets are 3D rather than nice flat sheets, you need to use something called the phase integral $q(\alpha)$ to work out the proportion of light that gets reflected towards the observer. Modelling your brown dwarf as a diffuse reflecting sphere (which is reasonable), you get a constant phase integral of ~0.41.

You can compute a planet's absolute magnitude $H$ using the formula $H = 5\log_{10}\frac{1329}{D\sqrt{p}}$ where $D$ is the object's diameter, and $p$ is its geometric albedo. For Jupiter, this gets you an absolute magnitude of ~9.5.

Throwing all our numbers together, you end up with your brown dwarf having an apparent magnitude of -6.67. This is pretty bright... brighter than any planet in the solar system seen from Earth (~5 times brighter than Venus), bright enough to see during daylight. It might even case faintly visible shadows at night time if your eyes are sharp enough. The moon is still >300 times brighter though.

The apparent angular diameter of your brown dwarf as seen from your planet is ~2'48" of arc. This is approximately the apparent angular diameter of the Mare Vaporum of the moon, which is here:

An image of the moon, with the Mare vaporum circled in red

(image source wikipedia, credit Gregory H. Revera)

You might be able to go an do a size comparison for yourself one night. This is large enough that your brown dwarf will be clearly visible as a blob, not a simple point of light, though do note that it won't appear to be circular because the phase angle means that only a portion of its surface will be brightly illuminated by its parent star. A good pair of binoculars should resolve large surface details if there are any (there may not be), and maybe even any large moons.

The "dark" side of your brown dwarf may or may not glow, depending on the age of the solar system and the amount of deuterium and helium-3 that it was formed with. Given its mass, it would probably be in Spectral class T, being heavy enough to burn deuterium but too light to burn lithium. A possible example of this sort of planet in real life might be Gliese 229B, a 40-60 jupiter-mass brown dwarf.

Alternatively, a younger world might be a lot more like Teide-1. It has a surface temperature over 2000K and a luminosity maybe as high as 0.0005 that of the Sun (but more on that in a moment). Unfortunately, Teide-1 is estimated to be 100 million years old or younger, and as such probably has a reasonable amount of its original supply of fusibles still remaining. 100 Myr isn't really very long by the standards of planetary evolution, let along the appearance of life (assuming the timescales of development of Earth are typical, of course). Gliese 229B may be as old as 3 billion years (according to Physical Properties of Gliese 229B Based on Newly Determined Carbon and Oxygen Abundances of Gliese 229A, arxiv) and as such is a better candidate for the appearance of your brown dwarf.

A more pessimistic take would end up with an old, cool brown dwarf with a surface temperature of <500K. This is also plausible, especially if your solar system is at least 4 billion years old, as ours is. Such a brown dwarf would not glow perceptibly.

A comparison of artists' renditions of brown dwarf stars with Jupiter, showing similar size but increasing amounts of luminosity as mass and fuel supply increase

(cropped from wikipedia, image credit MPIA/V. Joergens)

Finally (probably), luminosity. Teide-1 has a luminosity of 0.0005 that of the sun. Gliese 229B is much dimmer and cooler, and has a luminosity of 0.000011 times that of the sun. However, luminosity is the total amount of radiated energy, and as the Planck radiation formula tells us (in a complicated way you can gloss over for now) the actual spectrum of radiation emitted by a black body depends on its temperature. With a surface temperature of 2400K, Teide-1 only emits 4% of its energy in the visible spectrum. At only 950K, Gliese 229B emits less than 0.0001% of its energy as visible light.

Now, I'm less certain about turning these numbers into apparent magnitudes, so take these results with a bit of salt... my working suggests the brown dwarfs could be surprisingly bright which seem suspicious, but I don't have any better figures to go on right now. Teide-1 would be 50000 times dimmer than the sun, but that's still 8 times brighter than the full moon and quite bright enough to cast shadows at night. Gliese 229B by comparison would glow at magnitude -4.4... quite bright by the standards of our solar system, as bright as the brightest planets in our own sky, but isn't going to be casting shadows. The "dark" side would glow about 1/8th as bright as the sunlit side as seen from your trojan planet. The cool brown dwarf would not visibly glow.

All three dwarfs would produce a reasonable amount of near-IR light and be clearly visible in that spectrum. Teide-1 emit more than 30% of its energy output in the 650nm-1400nm band and would appear very bright, and even Gliese-229B emits 0.5% of its energy in that band. Longer wavelength IR is strongly absorbed by water and so would likely be less visible from the surface of your habitable trojan world. Night vision on your trojan world with one of the brighter dwarfs could be quite effective, though naturally evolving useful near-IR vision is likely to be challenging, because vision tends to rely on energetic short-wavelength photons with lazy long-wavelength IR photons getting literally lost in the noise.


Your question can be rephrased to "what would a brown dwarf look like as seen from a distance x, where x is the distance at which the observer is at the Lagrangian point of the brown dwarf-central star system?"

If I am not mistaken, L5 and the two other bodies form an equilateral triangle.

This means that the brown dwarf will be at 1.175 AU from the planet.

I cannot find any reliable source for estimating the size of a brown dwarf, but considering that we can see Jupiter from a larger distance than that, it will be for sure visible at night.

Visibility during the day might be limited to times around dusk and dawn, when the daylight is still not at full strength.

For an eye with spectral capabilities similar to the human eyes, the brown dwarf might mostly shine of reflected light, since its infrared emission would be invisible and its visible emission would be too faint when compared to the impinging starlight.


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