Astrophysics: Calculating the circumference of an ellipse

In summary, there were a few mistakes in the formula used to find the length of Earth's orbit around the sun without using Google. The formula should be sqrt(a^2-b^2) for the square root of the difference of squares, and {[(sqrt(a^2-b^2))/a]^4}/3 for the calculation of the length. Additionally, the semi-major and semi-minor axis were incorrectly used, as the semi-major axis is the average of the closest and farthest distances, and the semi-minor axis can be easily found using the eccentricity.
  • #1
Andy1200
2
0
Homework Statement
I'm new to this website. Could someone explain how to solve this equation, its the formula for an ellipse circumference :
Relevant Equations
P = 2a(pi){1-(1/2)^2[(sqrta^2b^2)/a]^2- [(1*3)/(2*4)]^2{[(sqrta^2b^2)/a]/3}^4.....}
Substituting :
a = (9.15x10^7 mi)+(9.45x10^7mi) = 1.86x10^8 mi
b = ( a/2 ) = 9.3x10^7 mi

For this, I used six terms and got :

1.075x10^9 miles

Is my math wrong?;
 
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  • #2
There seem to be a few mistakes in your formula. If I've got it right, "sqrta^2b^2" should be sqrt(a^2 - b^2). (Maybe just a typo on your part.)
{[(sqrta^2b^2)/a]/3}^4 should be {[(sqrt(a^2-b^2))/a]^4}/3
And I don't know where you're getting your a and b from. In this formula, a is the semi-major axis and b the semi-minor axis. Not (as you are using?) the sum and average of these. That's why you're getting an answer about a factor of 2 high.
 
  • #3
Thanks for the feedback. Yes, I'm still learning how to type formulas, but what you wrote is what I meant. I was using the formula to find the length of Earth's orbit around the sun w/out using Google. My apologies on the a and b numbers. I've found it extremely difficult to find the semi-minor axis of the orbit, given that the farthest and closest distances make a 180-degree angle.
 
  • #4
That's because the sun is at a focus of the ellipse, not the centre. In this case, the semi-major axis is the average of the closest and farthest distances. You can't get the semi-minor axis from these (it is independently variable), but you should be able to find it easily, or the value of the eccentricity, from which you can work it out, on the internet. (Actually the eccentricity is all you need for the circumference equation, if you have the semi-major axis.)
 

Related to Astrophysics: Calculating the circumference of an ellipse

1. How do you calculate the circumference of an ellipse?

The formula for calculating the circumference of an ellipse is: C = π(a + b) where a and b are the lengths of the semi-major and semi-minor axes respectively.

2. Why is calculating the circumference of an ellipse important in astrophysics?

Ellipses are a common shape in the orbits of celestial bodies, such as planets and moons. Calculating their circumference is important in determining the distance and speed of these bodies, which is crucial for understanding their motion and behavior.

3. How is an ellipse different from a circle in terms of calculating circumference?

An ellipse has two different radii (semi-major and semi-minor), while a circle has only one. This means that the formula for the circumference of an ellipse is different from that of a circle, which is simply C = 2πr.

4. Can the circumference of an ellipse be calculated using its eccentricity?

Yes, the eccentricity of an ellipse (e) can be used in the formula: C = π(a + b) * (1 + (3e^2 / 10 + √(4 - 3e^2)) where a and b are the semi-major and semi-minor axes.

5. Are there any other methods for calculating the circumference of an ellipse?

Yes, there are other methods such as using the parametric equation of an ellipse or numerical integration. However, the simplest and most commonly used method is the formula C = π(a + b).

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