Ellipse question....?

Question

I am totally lost on this problem. Could somebody please assist me? Thank you very much!

The planets move around the sun in elliptical orbits with the sun at one focus. The point in the orbit at which the planet is closest to the sun is called perihelion, and the point at which it is farthest is called aphelion. These points are the vertices of the orbit. The earth's distance from the sun is 147,000,000 km at perihelion and 153,000,000 km at aphelion. Find an equation for the earth's orbit. (Place the origin at the center of the orbit with the sun on the x-axis.)

Here's all I have so far:

(x^2 / a^2) + (y^2 / b^2) = 1

so...

(x^2 / 2.25 x 10^16) + y^2 / ???) = 1

Basically, I need help on the ??? part. Thanks again!

Answer 1

See the diagram at the link below.

a is the semi-major axis. The major axis is 2a, which is the sum of the aphelion and the perihelion.

a = (144 x 10^6 + 153 x 10^6) / 2 = 150 x 10^6 = 1.50 x 10^8

e is the eccentricity.

e^2 = 1 - (b^2 / a^2)
where b is the semi-minor axis.

From the diagram, the aphelion = ea + a
This gives e = 0.02

Substitute a and e into e^2 = 1 - (b^2 / a^2)

This gives b = 1.49994 x 10^8

Answer 2

Start with equation (x^2/a^2) + (y^2/b^2) = 1 like you did.
Find two points on the ellipse. Use the earth at perihelion and aphelion. Distance from earth to the center of the ellipse at either of these times is (P+A)/2

So two points on ellipse are ((P+A)/2 , 0) and (-(P+A)/2 ,0)
Put either of these points in for x and y and solve for a. You did this and you got a = 1.5 x 10^8 THAT IS CORRECT

You can not find "b" because an ellipse is not defined by just 2 points. Perihelion and aphelion does not give enough info. There are an infinite number of ellipses that pass thru these two points. The orbit could be a circle or it could be very long and thin. You would need to know its eccentricity or some other bit of info to completely define the equation.

You can let b equal any number except zero and you will have an equation for an ellipse with the given peri and ap distances.

I hope this helps you.
Jeff
[e-mail]

Answer 3

Aphelion is the distance a + c from the sun.
Perihelion is the distance a - c from the sun.

a + c = 153,000,000
a - c = 147,000,000

Adding we get:

2a = 300,000,000
a = 150,000,000
a = 1.5*10^8
a² = 2.25*10^16

a + c = 153,000,000
c = 153,000,000 - a = 153,000,000 - 150,000,000
c = 3,000,000
c = 3*10^6
c² = 9*10^12

b² = a² - c² = 2.25*10^16 - 9*10^12 = 10^12(2.25^10^4 - 9)

b² = 10^12(22,500 - 9) = 10^12(22,491) = 2.2491*10^16

If we place the center of the orbit at the origin we have the following equation of the earth's orbit:

x²/a² + y²/b² = 1

x² / (2.25*10^16) + y² / (2.2491*10^16) = 1