Comet C has an elliptical orbit of eccentricity 0.9638 with the sun at one focus. The greatest distance of the comet from the sun is 2940 million miles. Find the least distance between the comet C and the sun.
To find the least distance between the comet C and the Sun, we need to remember that the distance of a comet from the Sun can be calculated using the formula for the equation of an ellipse:
r = a(1 - e)
Where:
r = distance from the focus (Sun)
a = semi-major axis
e = eccentricity
Given that the greatest distance of the comet C from the Sun is 2940 million miles and the eccentricity is 0.9638, we can use this information to find the semi-major axis, as follows:
r = a(1 - e)
2940 = a(1 - 0.9638)
Next, we isolate a in the equation by dividing both sides by (1 - e):
2940 / (1 - 0.9638) = a
Now, let's calculate the value of a:
2940 / 0.0362 ≈ 81104.42
So, the semi-major axis (a) is approximately 81104.42 million miles.
Now, to calculate the least distance between the comet C and the Sun, we substitute the semi-major axis (a) and eccentricity (e) into the formula:
r = a(1 - e)
r = 81104.42(1 - 0.9638)
r ≈ 2901.93 million miles
Therefore, the least distance between the comet C and the Sun is approximately 2901.93 million miles.