This is probably more of a math question than astronomy, but I'd like to know the distance the Earth travels in one revolution around the sun (i.e. a year).

According to the data in Wikipedia, I think there may be several ways for me to calculate this but I'm not sure whether they are correct or how to do them:

1. Calculate by velocity and time: the average velocity is given as $107200 mathrmkm/h$ (a suspiciously round number; imprecise?) and one revolution takes $365.256363004$ days. That gives $textdistance = 107,200 mathrmkm/h times (365.256363004 times 24 mathrm h) = 939,731,570.736691 mathrmkm approx 939.731 mathrmGm$
   


2. Calculate by semi-major axis and eccentricity: my math is too weak for that. I searched the web and didn't find a solution for that (calculating the circumference of an ellipse exactly seems to be a problem?)


3. Any other way using the orbital data given in the Wikipedia article?

In essence, this can be made into a pretty generic question: How do I calculate the distance an object travels in one revolution using the usual orbital parameters?

You need to find the perimeter of an ellipse. This might be accurate enough for you (recalling that an ellipse is only an accurate description of an orbit for a two-body system.

An ellipse has no simple formula to find the perimeter. There appear to be several commonly used approximations based on infinite series (see for example "Maths is Fun").

e.g. Define
$$ h = frac(a-b)^2(a+b)^2,$$
where $a$ and $b$ are the semi-major and semi-minor axes and $b = asqrt1-e^2$, where $e$ is the eccentricity.

Then the result you want is
$$ p = pi(a+b)left[1 + frach4 + frach^264 + frach^3256 ...right]$$

All this is general, but might be overkill for your problem involving the Earth. If my calculator skills are working, then for the Earth's orbit around the Sun, $a = 1.00000011$ au, $e = 0.01671022$, $b = 0.99986048$ au and $h = 7times 10^-5$.

Using this and just the first 2 terms in the series, I get a perimeter of $1.999895pi$ au. So hardly a correction from $2pi a$.

However, if you were serious about accuracy then you would have to add to this about 12.5 orbits around the barycentre of the Earth-Moon system and a further correction for the motion of the Sun around the solar system barycentre.

I pondered this myself a couple of weeks ago. For Earth's orbital circumference Ramanujan's (q.v.) formula for the perimeter of an ellipse is good enough. (Actually, Jean Meeus says so in his book, but I can't find the reference. Sorry.) The Excel formula I used is:

=PI() * (F4 + F6) * (1 + 3 * ((F4 - F6)^2/(F4 + F6)^2)/(10 + (4 - 3 * ((F4 - F6)^2/(F4 + F6)^2))^1/2))

Where F4 and F6 are the semi-major and semi-minor axes, respectively.

Using this formula I get a circumference of 939.8855 X 10⁶ km.

The distance travelled by Earth in one revolution around the Sun as calculated by the data from Wikipedia's entry about Earth is about $936.013 Gm$ or $939.886Gm$, depending on the approach used for calculation. Since my math-fu is not very strong please point out any mistakes I've made.

Integral approach

It seems the exact way to calculate the perimeter/circumference is via the formula $C = 4aE(e)$ where $E$ is the complete elliptic integral of the second kind.

I've found a solution using Python. I tried using the numbers from Wikipedia here for consistency:

import numpy as np
from scipy.special import ellipe
a = 149598023000 # semi-major in meter
e = 0.0167086 # eccentricity
pe = 4 * a * ellipe(e)
print(pe) # 936013392131.0

So about $936.013 Gm$. That's quite off from the $approx 940Gm$ given by the other answers. I'm wondering why. I know the other methods are approximations but the error seems rather large to me.

Ramanujan's first approximation

Ramanujan has given two approximations. The first is $C approx pi[3(a+b) - sqrt10ab + 3(a^2+b^2) ]$. In Python:

import math
a = 149598023000
e = 0.0167086
b = a * math.sqrt(1 - e**2) # derive semi-minor
pe = math.pi * ( 3*(a+b) - math.sqrt( (3*a + b) * (a + 3*b) ) )
print(pe) # 9.39886493337e+11

Here we are at $939.886Gm$ which is closer to the solution from the other answers and the $velocity*time$ approach.

Ramanujan's second approximation

The formula used by @BillDOe is Ramanujan's second approximation: $C = pi(a+b)(1 + frac3h10 + sqrt4-3h)$ with $h = frac(a-b)^2(a+b)^2$. In Python:

a = 149598023000
e = 0.0167086
b = a * math.sqrt(1 - e**2) # derive semi-minor
h = (a-b)**2 / (a+b)**2
pe = math.pi * (a+b) * (1 + ((3*h) / (10 + math.sqrt(4 - 3*h))))
print(pe) # 9.39886493337e+11

That's the same result as the first approximation. The error for Ramanujan's approximations is stated as $h^3$ and $h^5$ respectively. That's about $1.157^-25$ and $2.747^-42$. I'm not a math geek, so these sound pretty low for me and I wonder why the solution is so different than the integral approach.

@RobJeffries's approach

I don't know how this is called. The formula is:

$$ C = pi(a+b)left[1 + frach4 + frach^264 + frach^3256 ...right]$$

In Python:

a = 149598023000
e = 0.0167086
b = a * math.sqrt(1 - e**2) # derive semi-minor
h = (a-b)**2 / (a+b)**2
pe = math.pi * (a+b) * (1 + (h/4) + (h**2/64) + (h**3/256))
print(pe) # 9.39886493337e+11

Again, $939.886Gm$