Mercury is closer to Earth, on average, than Venus

Apologies for blowing my own trumpet, but after I listened to the episode I was so captivated by the idea that I tried my hand at visualising it: https://flother.is/2019/which-planet-is-closest-to-earth/

Oliver Hawkins, the researcher who found the answer for More or Less in the first place, also wrote about it: https://olihawkins.com/2019/02/1

Closest approach to Earth is of course what NASA cares about since they plan trips to it. I wonder what future the writer is imagining where average distance matters.

Hoo, boy

Also: If your wonderful new metric gives every planet the SAME closest neighbor, then maybe its not a meaningful metric.

    import numpy as np
    from scipy.integrate import dblquad
    
    pi = np.pi

    def av_distance(r1, r2):
        def distance(theta1, theta2):
            z1 = r1 * np.exp(1j * theta1)
            z2 = r2 * np.exp(1j * theta2)
            return abs(z2 - z1)
        integral, err = dblquad(distance, 0, 2*pi, 0, 2*pi)
        return 1 / (4 * pi ** 2) * integral

    print(av_distance(1, 0.723332))
    print(av_distance(1, 0.387098))

    1.063544409973366
    1.0156870128527526

I used to wonder about the distances if the planetary orbits were not in a plane (which also always seemed weird to me as a kid) and even though I utterly lacked the math to figure it out it was easy to develop an "in principle" qualitative understanding.

School seems to be oriented backwards.

From the table at the end, it also turns out that the Sun (at ~1AU) is even closer to Earth, on average, than Mercury.

(The Sun is almost certainly average-closest for Mercury itself, but unsure if the Sun would be average-closest for Venus – the table doesn't explicitly include the Sun.)

             _.-""""-._
           .'   ___    `.
          /   .' __ `.   \
         |   /  /  \  \   |
         E   |  |  M  V   |
         |    \  --  /    |
          \    `.___'    /
           `._        _.'
              `-....-'

In other words, Mercury is on average closer to Neptune than, say, Saturn is, and indeed Mercury is on average closer to every planet than any other planet is.

This is both somewhat surprising (I'd have thought that there would be a symmetry here - that the average distance between Neptune and Saturn would be the same as the average distance between Neptune and the Sun), and also (I think) uninformative.

The way orbital dynamicists think about "close" orbits is in terms of delta-V, which is to say, how big a change in velocity does it take to change one orbit to a different one. Or, in more concrete terms, how much fuel would it take for a spacecraft to travel from a body in one orbit to a body in a different orbit. That's a completely different way to think about closeness than the OP 's metric.

Spaceflight is non-Euclidean like that.

The OP's metric would be helpful for estimating average time-of-flight communications delay, but over distances like this I don't think that averages are that informative. The total variation is quite large; basically it varies as |r1|-|r2| to |r1|+|r2|. For Earth-Mars that's from 4.5 to 20 minutes. Not sure knowing the "average" distance is all that helpful there.

For Mars-Neptune it's 4 hours to 4.4 hours, and for Earth-Neptune it's 4 to 4.3 hours. Even though the distance is much greater, the orbital radius of the outer planets is so large that the variation as the inner planet goes around the sun is almost in the noise.

Takeaway: for comms delay to the the outer planets, just calculate the time-of-flight distance to the sun.

Though thinking about it a space station ring or something orbiting around that gravity anchor might be better in some aspects.

Wouldn't the communication relay depend on just the farthest distance, rather than the average distance?