# Distance to Sun at Perihilion and Aphelion at max and min Eccentricity

Title says it all. I'm putting together a presentation on Milankovitch Cycles, and I would like to bring up the difference in distance to the sun (km) at aphelion and perihelion when the earths orbit is at maximum and minimum eccentricity. The numbers I've found are that the maximum eccentricity of Earth's orbit is 0.0679, and the minimum is 0.000055.

I like the math of ellipses cause there's enough parts to keep it interesting, but it's not super complex either. The answer to your question as you put it, is remarkably simple. I've bolded the letters that represent distances for easier reading.

Eccentricity or $e = c/a$, where $a$ is the semi major axis and $c$ is the distance between the focal point (sun's location) and the center of the ellipse. $ea$ is the distance between the center point and the focal point.

So, the answer to your question is simply $a$ plus $ea$ and $a$ minus $ea$. For the Earth and sun, the semi major axis or $a$, is $149,600,000\:\mathrm{km}$, then simply add and subtract $ea$ to both.

When $e = .0679$, $ea = 10.15\:\mathrm{million\:km}$. So $149,600,000 \pm 10,150,000$. $159,750,000$ and $139,450,000\:\mathrm{km}$ at furthest and closest.

When $e = .000055$, $ea = .008\:\mathrm{million\:km}$, so $149,680,000$ and $149,520,000$ at most circular.

Simple picture

More detailed picture

Maths and sources of pictures, here and here

footnote: I hope this wasn't a homework question.

• I guess in a sense it's homework for my job as a guide. I'm putting together a presentation about ice ages, and I'm talking about Milankovitch Cycles, but I wanted to include this info just so people could conceptualize the variances better. – Fred Oct 20 '16 at 23:08
• Converted your bold math text to latex, feel free to undo if you want. – zephyr Oct 21 '16 at 12:45

I'm not sure where in the above, the eccentricity of 0.0679 came from, maybe he meant 0.0167. Also, the Earths orbit does not come that close to the sun as the diagram shows. The orbit is near circular, with an eccentricity of 0.01671123, which means a and b (which isn't show in the above) are very close values to one another. The Focus points can be given be $F_1 = (-ae, 0)$ and $F_2 = (ae, 0)$ and eccentricity being $e = \sqrt{1-(b^2/a^2)}$. Use Pythagoras to figure the rest out.

• I've seen the numbers before for high and low, so I didn't even question that a source wasn't given. Here's a common source for those two numbers: en.wikipedia.org/wiki/… – userLTK May 8 '18 at 2:50
• The Milankovitch hypothesis is a suggestion on previous states of the Earth's orbits, but not today. I don't personally know much about the validity of the hypothesis. Wikipedia has the eccentricity document as 0.0167086, which is close to the value I calculated. These two number are a few significant digits apart, but close enough. I would probably use their number, not the one I calculated. I'm sure a professional astronomer made the calculation with much better data than I have. – Byron May 9 '18 at 4:56
• I think Milankovich cycles are much more than a hypothesis. They're well established. The math gets a bit tricky, but future and past eccentricities can be calculated easily enough with today's computers and perturbation theory mathematics. More details here: en.wikipedia.org/wiki/Perturbation_(astronomy)#Periodic_nature and scholarpedia.org/article/… As for the precision of the numbers, I wouldn't bet my savings on any number Wikipedia provides, but they're probably in the ballpark. – userLTK May 9 '18 at 14:58
• I don't know. The Solar System's periods are stable on human terms. If you go out millions of years, there is no telling what happens. Though, I would point out that Charles Hapgood came up with a theory relative to the the icecaps shifting and Einstein rebuked the hypothesis. There have been many other hypothesis made about climate, most recently, CO2 emission are the cause of all climate change. I prefer to live in the here and now. – Byron May 10 '18 at 20:13