Use the data from the diagram to calculate a, b, and c of this elliptical orbit.

Question

Mars is 207,000,000 km from the Sun at its closest point. Earth's orbit is almost circular, with an eccentricity of only 0.0167. The eccentricity of Earth's orbit is 0.0934. Mars is 249,000,000 km from the Sun at its farthest point.

1. Mars travels in an elliptical orbit with the Sun at one of its foci. Use the data from the diagram to calculate a, b, and c of this elliptical orbit. 

2. Point P is the midpoint between Mars’ closest and farthest distances to the Sun. Use your values of a and b to write an equation of the elliptical orbit of Mars relative to a coordinate system drawn through point P (y-axis in gray). Use distances in millions of kilometers.

3. It is also useful to define Mars’ motion relative to the Sun. Imagine a new coordinate system (y-axis in black) with its origin at the center of the Sun. Rewrite your equation for the ellipse in this new coordinate system

4. Explain how the eccentricity of a planet’s orbit can affect its annual weather cycle.

Answer 1

1. The equation of an ellipse in Cartesian coordinates is:

x²/a²+y²/b²=1 where a and b are the lengths of the semi-major and semi-minor axes (also known as the orthogonal radii). c=√(a²-b²) and c/a or √(1-b²/a²) is the eccentricity e. c is the distance between the centre of the ellipse and its focus.

Aphelion is the point on the orbit furthest from the sun while perihelion is that nearest to the sun. In an elliptical orbit, the sum k of the distances between the point on the orbit and each of its foci has to be constant, so if P(-a,0) or P(a,0), the geometry shows that k=a-c+a+c=2a. For the point P(x,y) anywhere on the orbit, the sum of the distances from P to each of the foci must also be 2a. If the sun is at (c,0), then aphelion is at (-a,0) (distance a+c from the sun) and perihelion at (a,0) (distance a-c from the sun). If we take either of these points on the orbit we get k=2a, which is also the sum of the two distances. Therefore:

2a=249M+207M=456M, a=228M km. M=1,000,000 km. (In other words, a is the average of the two distances.)

e=c/a=√(1-b²/a²), e²=1-b²/a², b²=a²(1-e²)=228M²(1-0.0167²)⇒b~228M km. (b=227,968,000 more accurately.)

a=228,000,000 km, b=227,968,000km approx,

c²=a²-b²=(a-b)(a+b)=32000×455968000=1.449782×10¹³, c=3,807,600km approx. (Some figures have been rounded for convenience.)

2. No picture has been provided.

If the sun is at (c,0), one of the foci, then perihelion is (a,0) and aphelion is (-a,0) so the midpoint is (0,0) and the equation is the standard ellipse x²/a²+y²/b²=1, where a and b are as calculated earlier.

3. No picture has been provided.

Displacement (horizontal shift) of the equation to a focus changes the equation to:

(x-c)²/a²+y²/b²=1 when the sun is the centre of the ellipse.

4. At perihelion the planet will be warmest and at aphelion coolest. Between aphelion and perihelion the planet experiences spring (as it moves from aphelion to perihelion) and autumn or fall (as it moves from perihelion to aphelion). A low eccentricity (e→0) means that there is very little change in the weather throughout the year. A high eccentricity produces extreme differences in the weather. The planet's rotation and axial tilt also have profound affects on weather. A captured orbit, where one hemisphere perpetually faces the sun greatly exaggerates the extremes.