Is it possible to calculate how much a star's brightness dims when a planet transits in front of it from a viewer's perspective?

Question

Last night I got to thinking about what would happen if Jupiter and Venus suddenly switched places.

Since Venus comes the closest to Earth than any other planet and Jupiter is much larger, does that mean that Jupiter (now following the orbit of Venus) would appear large enough to eclipse the Sun from Earth's perspective?

Jupiter's angular diameter in this scenario is given by $$\delta = 2\arctan \left(\frac{d}{2D}\right)$$

where

• $\delta$ is its angular diameter,

• $d$ is its actual diameter,

• $D$ is the distance between both bodies.

The closest Venus and Earth ever get to each other (in the near future) is $d = 39.5 \times 10^6$ km, and Jupiter's diameter is $D = 139,820$ km.

So if Jupiter replaced Venus, its maximum angular diameter would be $$\delta = 2\arctan \left(\frac{139,820}{2 \times 39.5 \times 10^6}\right)$$ $$\delta = 0.202812°, or \ 0° 12 ' 10.12 " $$

The Sun's angular diameter as seen from Earth is about 30 arcminutes, and so Jupiter in this scenario with only 12 arcminutes would not appear to be large enough to completely eclipse the Sun.

However, assuming it is a total transit, it would still cover $≈16.27 \%$ of the Sun's disk as seen from Earth. Surely that would result in a noticeable dim in the brightness of the Sun as seen from Earth, no?

Is there an equation to calculate how much a star's brightness dims when a planet transits in front of it from a viewer's perspective?

Answer 1

Yes! This type of reasoning, called transit photometry, is used frequently in the study of exoplanets. The Kepler mission was designed primarily to identify exoplanets using this method, by identifying stars which exhibit the kind of periodic fluctuations in brightness that can be associated with the periodic occlusion of the star by an orbiting planet.

Calculating the precise amount a star's brightness dims from an occluding planet can be tricky. Stars don't have a perfectly uniform luminosity (thanks to limb darkening) and planets aren't perfectly spherical, so the luminosity will depend on exactly where the transit is occurring relative to the star.

Putting these limitations aside, though, you can get a pretty decent estimate of how much a star's luminosity drops from a fully transiting planet simply by looking at the ratio of the star's surface area that's covered from the perspective of the viewer. Let's model the Sun naively as a 2D circle emitting light directly towards Earth. When 16% of the Sun is covered by a Jupiter transit like you described, only 84% of the light from the Sun would still reach the Earth. This, in turn, means that the luminosity of the sun would be 84% of the pre-transit value.