# How would rainbows appear on other planets?

Are other planets capable of producing rainbows? How would those rainbows appear? Can rain, clouds or ice from elements other than water produce rainbows?

note 1: I've verified @JamesK's answer's index of refraction of 1.27 (since no source was cited), at least for a temperature of 111K, yay! On a colder day, say 90K, the index goes up and the rainbow will shrink a few degrees, close to the size of that on Earth.

Source for methane:

Source for water:

Now @CarlWitthoft shows two unlabeled plots with no sources cited and very different values for $$n$$.

note 2: @CarlWitthoft's unsourced claim that methane has a significantly lower dispersion than water in visible light appears to be without merit. I've plotted both materials on the same axis and they are comparable. The rainbows will have slightly different spreading of colors, but I do not thing the rainbow will disappoint!

@JamesK's answer mentions that Titan could see rainbows from liquid methane rain.

Using math from 1, 2, 3:

$$k = \frac{n_{droplet}}{n_{atmosphere}}$$

$$\alpha = \arcsin\left(\sqrt{ \frac{r-k^2}{3} } \right)$$ $$\beta = \arcsin\left( \frac{\sin\alpha}{k} \right)$$ $$\theta = 2\phi = 4\beta - 2\arcsin(k \sin \beta)$$

Actually, lower index makes the rainbow larger. Remember that red is on the outside. With $$k=4/3\approx1.33$$ the rainbow is at ~42°, for $$k=1.27$$ it blows up to ~52°.

All else equal it would be a little brighter as well; with a larger incident angle at the back of the drop, the fresnel reflection will be a bit stronger.

Source

# https://www.stewartcalculus.com/data/ESSENTIAL%20CALCULUS%202e/upfiles/instructor/eclt_wp_0301_inst.pdf

import numpy as np
import matplotlib.pyplot as plt

halfpi, pi, twopi = [f*np.pi for f in (0.5, 1, 2)]

k = np.linspace(1.2, 1.5, 31)

alpha = np.arcsin(np.sqrt((4.-k**2)/3.))
beta  = np.arcsin(np.sin(alpha)/k)
phi   = 2*beta - np.arcsin(k*np.sin(beta))
theta = 2 * phi

things = (alpha, beta, theta)
names  = ('alpha', 'beta', 'theta = 2phi')
if True:
plt.figure()
for i, (thing, name) in enumerate(zip(things, names)):
plt.subplot(3, 1, i+1)
plt.plot(k, degs*thing)
plt.title(name, fontsize=16)
plt.plot(k[7],  degs*thing[7],  'ok')
plt.plot(k[13], degs*thing[13], 'ok')
plt.show()

• I can't tell but I think you've missed the important part: water is dispersive (delta $n$ with $\lambda$ ; if methane is not then all wavelengths enter and exit at the same angle, and no rainbow. – Carl Witthoft Feb 22 '19 at 16:40
• @CarlWitthoft "...if methane is not (dispersive)..." can you name even one dielectric that isn't? Dispersion in visible wavelengths comes from absorption in the UV and is a pretty universal attribute of collections of atoms. I think you mean "substantially less dispersive than water" – uhoh Feb 22 '19 at 23:04
• Regarding the refractive index of methane, this may be of use (pdf) – user24157 Feb 22 '19 at 23:48
• @mistertribs thank you very much; I've incorporated that into my answer. – uhoh Feb 23 '19 at 11:41

Rainbows occur when sunlight shines through rain. This is rare in the solar system. Rain (of sulphuric acid) might be common enough under Venus's clouds, but there is no sun. Conversely, there is plenty of sun in Mars, but no rain, and only very rare clouds.

It rains on Titan: methane rain. Methane has a lower refractive index than water (1.27 instead of 1.33), which would make the rainbows slightly larger (though not by much 42->52). However the atmosphere of Titan is hazy, and while there is some light on the surface, the sun's disc is not visible.

There is rain in some layers of the gas giants, but again not on the outer layers where the sun is visible.

It is likely that the Earth is the only place in the solar system where rainbows are a common phenomenon.

• Maybe they are there but we can't see them because the sun, planets outside Earth's orbit and observer is never around that 40degree angle needed to produce a rainbow from the Sun of the atmosphere. – Muze Feb 21 '19 at 21:07
• Yes. Earth should be the only place where rainbows are vulgar. Other celestrial bodies should be also able to support rainbows where there is mist or vapor of some chemical, and enough sunlight, but those criteria are rarely met. – Max0815 Feb 22 '19 at 2:56
• It's not the refractive index which leads to rainbows but rather the dispersion (variation of $n$ with wavelength). – Carl Witthoft Feb 22 '19 at 16:38
• @CarlWitthoft When dispersion is low (or the spreading is otherwise confounded) there will still be a rainbow, but it will be less colorful; it may stop dispersing but it doesn't stop refracting! See What actually happens to reduce the perceived color in a 'white rainbow“ or ”fog-bow"? – uhoh Feb 23 '19 at 0:14
• What do you mean by "but there is no sun (in Venus)"? – Nilay Ghosh May 17 at 5:22

Take a look at these charts. The methane one is the best I could find on a quick search, but it suggests the dispersion over the visible wavelength band is a fraction of the value for water.

Since the existence of a rainbow depends on the ability of the substance to 'bend' different wavelengths different amounts, you can see that methane, at least, would produce a rather unsatisfying rainbow. And even that assumes that you had an atmosphere which supported methane droplets of an appropriate size to achieve a prismatic effect.

Roughly speaking, you would want the methane droplets to be larger than the water droplets which produce rainbows on Earth by the ratio of their dispersions. This is because the angular output spread depends in part on the length of path thru the droplets.

• Any differences in the range of color in the rainbow? Keep in mind not only the form of rain can produce rainbow. Clouds of Jupiter and other planets can as well. – Muze Feb 22 '19 at 19:43
• @Muze Unless the molecule in question (water, methane or other) has a severely sharp absorption edge, the color range is limited only by our retinal ability to discriminate wavelengths. – Carl Witthoft Feb 22 '19 at 19:44
• Yes but doesn't most transparent liquids refract light? – Muze Feb 22 '19 at 19:46
• @Muze there's two things here that often get lumped together, and they shouldn't be. While refract just means bend, disperse means bend different colors differently. If you had rain droplets (or prisms) with low dispersion you would still get a rainbow, but it would be white. What actually happens to reduce the perceived color in a 'white rainbow“ or ”fog-bow"? Carl and many others might be "unsatisfied" by it, but it would still be there, narrower and more concentrated but less colorful. – uhoh Feb 22 '19 at 23:58
• @uhoh yeah you're partly right - the angular output (not just the translation) depends on the entrance and exit angles more than the droplet size. – Carl Witthoft Feb 25 '19 at 12:30