How would rainbows appear on other planets?

Question

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

Related: https://space.stackexchange.com/questions/34357/rainbow-space-probe

Answer 1

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:

• https://refractiveindex.info/?shelf=organic&book=methane&page=Martonchik-liquid-111K which comes from
• Optical constants of liquid and solid methane John V. Martonchik and Glenn S. Orton, Applied Optics Vol. 33, Issue 36, pp. 8306-8317 (1994) https://doi.org/10.1364/AO.33.008306
• available at ResearchGate
• and through NASA thanks to @mistertribs's comment

Source for water:

• https://www.osapublishing.org/ao/abstract.cfm?URI=ao-12-3-555
• Optical Constants of Water in the 200-nm to 200-Mm Wavelength Region
• George M. Hale and Marvin R. Querry March 1973 / Vol. 12, No. 3 / APPLIED OPTICS 555

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!

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@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
# https://www.physics.harvard.edu/uploads/files/undergrad/probweek/sol81.pdf
# nice math http://www.trishock.com/academic/rainbows.shtml

import numpy as np
import matplotlib.pyplot as plt

halfpi, pi, twopi = [f*np.pi for f in (0.5, 1, 2)]
degs, rads = 180/pi, pi/180

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()

Answer 2

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.

Answer 3

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.

Answer 4

Somewhat outside the box is the Jovian moon Io. Direct sunlight us rwadily availabke, albeit only about 4% as ibtense at it would be at Earth's distance. Instead of rain, Io produces sulfur-bearung volcanic gases that condense on the cold environment of the space surrounding Io.

That the sulfurous compounds can disperse light is suggested by [Maurizio and Aschauer 1](https://doi.org/10.1088/0957-0233/11/12/310), who show that sulfur dioxide is about 2% more refractive to violet light than to orange light. The chief limitation appears to be whether the density of the material would be sufficient to make a visible bow.

Source: NASA/Johns Hopkins University Applied Physics Laboratory/Southwest Research Institute. Retrieved from space.com.

Reference

1. Musso, Maurizio & Aschauer, R & Asenbaum, A. & Vasi, Cirino & Wilhelm, Emmerich. (2000). "Interferometric determination of the refractive index of liquid sulphur dioxide". Measurement Science and Technology. 11. 1714. 10.1088/0957-0233/11/12/310.