## TL;DR
Contrary to popular belief, Venus is not Earth's closes...

CGP Grey did an [interesting video](https://www.youtube.com/watch?v...

![](https://i.imgur.com/UxUaFj9.png)
This diagram illustrates the ...

Could you provide a link to elliptical integrals, generally, and on...

![](https://www.flother.is/processed_images/ee27b186415fe87e00.png)...

Here's the link to the BBC episode of [More or Less](https://www.bb...

It should say Figure 4 instead of Figure 3 in comparing the results...

Venus is not Earth’s closest neighbor

Tom Stockman, Gabriel Monroe, and Samuel Cordner

12 March 2019

Abstract

Calculations and simulations conﬁrm that on average, Mercury is the

nearest planet to Earth—and to every other planet in the solar system.

Figure 1: An illustration of the solar system. Credit: NASA

1 Introduction

Which planet is closest to Earth? Ask an astronomer or a search engine, and

you’ll probably hear that though the situation changes frequently, Venus is

the closest when averaged over time. Several educational websites, such as The

Planets and Space Dictionary, publish the distance between each pair of planets,

and they all show that Venus is nearest to Earth on average. They’re all wrong.

NASA literature even tells us Venus is “our closest planetary neighbor,” which

is true if we are talking about which planet has the closest approach to Earth

but not if we want to know which planet is closest on average.

As it turns out, by some phenomenon of carelessness, ambiguity, or group-

think, science popularizers have disseminated information based on a ﬂawed

assumption about the average distance between planets. Using a mathematical

method that we devised, we determine that when averaged over time, Earth’s

nearest neighbor is in fact Mercury.

1

2 Simple but wrong

To calculate the average distance between two planets, The Planets and other

websites assume the orbits are coplanar and subtract the average radius of the

inner orbit, r

1

, from the average radius of the outer orbit, r

2

. The distance

between Earth (1 astronomical unit from the Sun) and Venus (0.72 AU) comes

out to 0.28 AU.

Although it feels intuitive that the average distance between every point

on two concentric ellipses would be the diﬀerence in their radii, in reality that

diﬀerence determines only the average distance of the ellipses’ closest points.

3 A better approach

To more accurately capture the average distance between planets, we devised

the point-circle method (PCM). The PCM treats the orbits of two objects as

circular, concentric, and coplanar. An object in a circular orbit maintains con-

stant velocity, which means that over a suﬃciently long period, it is equally

likely to be in any position in that orbit.

Figure 2: (a) The uniform probabilistic distribution of two circularly orbiting

bodies, with orbital radii r

1

and r

2

. (b) Radial symmetry is used to simplify

the average distance between the distributions.

It can be determined by integrating the point-to-point distance around c

1

,

as simpliﬁed in the following equation, which deﬁnes the PCM:

¯

d =

2

π

(r

1

+ r

2

)E

2

√

r

1

r

2

r

1

+ r

2

(1)

where E(x) is an elliptic integral of the second kind. The average distances

between pairs of planets determined by the PCM are included in the table.

Venus is on average 1.14 AU from Earth, but Mercury is a much closer 1.04 AU.

From the PCM, we noticed that the distance between two orbiting bodies is

at a minimum when the inner orbit is at a minimum. That observation results in

2

Figure 3: A simulation of an Earth year’s worth of orbits by the terrestrial

planets begins to reveal that Mercury (gray in orbital animation) has the small-

est average distance from Earth (blue) and is most frequently Earth’s nearest

neighbor. A longer run of the simulation can be seen on YouTube. In addition,

planetary geoscientist David Rothery ran a solar system simulation for the BBC

radio program More or Less and came up with similar results.

what we call the whirly-dirly corollary (named after an episode of the cartoon

Rick and Morty): For two bodies with roughly coplanar, concentric, circular

orbits, the average distance between the two bodies decreases as the radius of

the inner orbit decreases. It’s clear from this corollary, and from the table,

that Mercury (average orbital radius of 0.39 AU), not Venus (average radius of

0.72 AU), is the closest planet to Earth on average. In fact, Mercury is even

the closest planet to Neptune. (And yes, to Pluto too: Though the corollary

doesn’t work as well for the dwarf planet, with its orbital inclination of 17° and

eccentricity of 0.25, its nearest neighbor is also Mercury.)

4 Simulation validation

We ran a simulation to conﬁrm the whirly-dirly corollary, using a Python library

called PyEphem to chart the positions of all eight planets in the solar system

for 10 000 years. An animation illustrating the simulation is shown in ﬁgure

2. After each 24 hours of simulated time, the program records the distances

between each pair of planets.

In the table we list the average measured distances over the 10 millennia

and compare them with the results from the PCM and the traditional method.

The simulation results diﬀer from the ﬂawed numbers by up to 300 %, but

they deviate from the PCM ﬁgures by less than 1 %. Figure 3 compares the

results from both methods with the simulation of the average distance between

Neptune and the seven other planets.

3

Figure 4: The point-circle method (blue points) calculates the average dis-

tance from Neptune to the other seven planets far more accurately than the

method used by websites such as The Planets (yellow). The values are com-

pared with those obtained from the 10000-year solar system simulation. Credit:

Greg Stasiewicz and Flourish

5 Conclusion

As best we can tell, no one has come up with a concept like PCM to compare

orbits. With the right assumptions, PCM could possibly be used to get a quick

estimate of the average distance between any set of orbiting bodies. Perhaps it

can be useful for quickly estimating satellite communication relays, for which

signal strength falls oﬀ with the square of distance. In any case, at least we know

now that Venus is not our closest neighbor—and that Mercury is everybody’s.

4

Figure 5: A comparison of the average distances (in astronomical units) between

planets, as obtained via simulation, the point-circle method, and the common

method used by various educational websites. The PCM results are far closer to

those from the simulation, correctly demonstrating that Mercury is every other

planet’s closest neighbor.

5

## TL;DR
Contrary to popular belief, Venus is not Earth's closest neighbor on average. This misunderstanding stems from a common but flawed assumption about the average distance between planets. Averaged over time, Mercury is actually the closest planet to Earth — and to every other planet in the solar system. This new approach can potentially improve our understanding of orbital dynamics and could be useful in estimating satellite communication relays.
![](https://www.flother.is/processed_images/ee27b186415fe87e00.png)
You can actually use this [python script](https://github.com/flother/nearest-planet) by Matt Rigott to calculate which planet will be our nearest neighbour over the coming months and years!
CGP Grey did an [interesting video](https://www.youtube.com/watch?v=SumDHcnCRuU) on this topic.
Here's the link to the BBC episode of [More or Less](https://www.bbc.co.uk/programmes/m0001y9p).
![](https://i.imgur.com/UxUaFj9.png)
This diagram illustrates the terms used in the point-circle problem.
The equation for distance in a radial coordinate system is
$$
\text {d}=\sqrt{r_1^2+r_2^2+r_1 r_2 \cos (\theta)}
$$
The equation for finding the average value of a function is
$$
\text { Average Value }=\frac{1}{b-a} \int_a^b f(x) d x
$$
Substituting $f(x)$ with Distance $(\theta)$ from the previous equation and integrating from 0 to $2 \pi$ yields
$$
\text { Average Distance }=\frac{1}{2 \pi} \int_0^{2 \pi} \sqrt{\left(r_1^2+r_2^2+r_1 r_2 \cos (\theta)\right.} d \theta
$$
which can be simplified using a complete elliptic integral of the second kind.
$$
\text { Average Distance }=\frac{2}{\pi}\left(r_1+r_2\right) E\left(\frac{2 \sqrt{r_1 r_2}}{r_1+r_2}\right)
$$
where $E()$ is an elliptic integral of the second kind. The use of this last equation for calculating the average distance between 2 orbiting bodies will hence be referred to as the Point-Circle Method (PCM).
The average distances between pairs of planets determined by the PCM are shown as part of Figure 6.
The Whirly Dirly Corollary
Plotting the last equation reveals an interesting behavior. The distance between two orbiting bodies is at a minimum when the inner orbit is at a minimum. This observation results in the Whirly Dirly Corollary (WDC): for two bodies with roughly concentric circular orbits around a common center, the average distance between the two bodies decreases as the radius of the inner orbit decreases. From this corollary it is observed that Mercury (average orbital distance of 0.39 AU), not Venus (average orbital distance of $0.72 \mathrm{AU}$ ), is the closest planet to Earth on average.
It should say Figure 4 instead of Figure 3 in comparing the results from both methods with the simulation of the average distance between Neptune and the seven other planets.
Could you provide a link to elliptical integrals, generally, and one for elliptic integrals of the second kind, so that we could learn how equation (1) was obtained. Thank you.