+0

0
21
1
+124

The planets in our solar system do not travel in circular paths. Rather, their orbits are elliptical. The Sun is located at a focus of the ellipse.

The perihelion is the point in a planet’s orbit that is closest to the Sun. So, it is the endpoint of the major axis that is closest to the Sun.

The aphelion is the point in the planet’s orbit that is furthest from the Sun. So, it is the endpoint of the major axis that is furthest from the Sun.

The closest Mercury comes to the Sun is about 46 million kilometers. The farthest Mercury travels from the Sun is about 70 million kilometers.

1.      What is the eccentricity of the ellipse? Round your answer to the nearest thousandth.

Sep 24, 2023

#1
+128732
+1

The major axis length =  46 +  70  = 116 =  2a

So a = the semi-major axis length =   116 / 2 =  58  =  a

The focal length =   58  - 46  =  12  =    c

We can find the  length of the semi-minor axis as

sqrt [ a^2 -c^2 ] = sqrt [ 58^2 - 12^2]  = sqrt [3220] = b   →   b^2  = 3220

The eccentricity is  given by

sqrt  [  1 -  b^2 / a^2 ] =

sqrt [ 1  -   3220 / 58^2 ] =

sqrt [ 1 - 805/841  ] =

sqrt [ 36 / 841 ]  =

6 / 29    ≈     .21

Sep 25, 2023
edited by CPhill  Sep 25, 2023