# Exactly how elliptical is Mercury's orbit, visually, without exaggeration?

First the full question: What does the orbit of Mercury really look like visually, without artistic liberties taken to amplify the eccentricity of the orbit, or simplifying it to appear as a perfect circle? What eccentricities are needed for orbits that readily appear elliptical?

I originally did this as a self-answer, but my answer was of such poor quality I've now deleted it. I hope to do a better answer backed by some calculations, but, as always, anyone else is free to answer.

• Strange wording: the orbit just is , and "visually" really has nothing to do with it -- other than the angle of observation. – Carl Witthoft Nov 23 '16 at 13:10

Some things we know about Mercury's orbit:

• Semi-major axis: 0.387 AU, about 57.9 million km
• Eccentricity: 0.205

We can calculate the semi-minor axis, $b$ from the equation $$e=\sqrt{1-\frac{b^2}{a^2}}\to b=0.379\text{ AU}$$ We can also calculate the distance to the focus from the center of the ellipse, $f$, as $$f=ae=0.078\text{ AU}$$ I used Mathematica's ParametricPlot to plot Mercury's orbit using these parameters, along with Earth's. Earth's semi-major axis is 1 AU, and with an eccentricity of 0.016, its semi-minor axis is about 1 to three significant figures; I've treated it as a circle. Mathematica gives

ParametricPlot[{{0.387*Sin[x] - 0.0793, 0.379*Cos[x]}, {Sin[x],
Cos[x]}}, {x, 0, 2*Pi}, AxesLabel -> {AU, AU}, PlotLegends ->
Placed[LineLegend[{Red, Blue}, {"Mercury", "Earth"}], {1, 0.25}],
PlotStyle -> {Red, Blue}]


I'd consider the deviation from a perfect circle to be noticeable insofar as the focus is shifted from the point $(0, 0)$, but it's not incredible.

• I would argue that it's not noticeable and that the primary effect of the eccentricity is that the Sun is noticeably not the center of the orbit. – user21 Nov 22 '16 at 20:59
• @barrycarter That's basically what I meant, but I can make that clearer. – HDE 226868 Nov 22 '16 at 21:00
• Looks OK compared with the NASA version. – Rob Jeffries Nov 22 '16 at 21:29

Here you go. Image is from this NASA website and shows Earth, Mercury and Mars, with circles for comparison.

@rob-jeffries and @hde-226868 have both given excellent answers: after remembering that I was the OP, I upvoted both and approved one.This answer is mostly my attempt at redemption:

The image above (Sun depicted at 3 times actual diameter) shows the orbit of Mercury with the Sun as the center. This agrees with two other images that are also meant to show Mercury's orbit to scale:

Both of these look similar to my answer, so I'm reasonably confident that I'm correct this time. https://github.com/barrycarter/bcapps/blob/master/STACK/bc-mercury.m if anyone wants to check my work

Things to note:

• The primary effect of the eccentricity is that the center of Mercury's orbit is quite far from the Sun.

• The orbit's center is about 0.0796 AU or 11.9 million km from the Sun, and the other focus is 23.8 million km from the Sun.

• Mercury's semiminor axis is 97.86% the length of its semimajor axis, so is looks very much like a circle. To see this, we plot again, this time using the midpoint of the two foci as the origin:

The x diameter is 587 pixels and the y diameter is 575 pixels. If you draw a blue circle with a radius averaging Mercury's semiminor and semimajor axes, you get:

I'll leave it to the reader to decide if Mercury's orbit does or does not look like a circle.