Question

# The orbit of the comet Encke has a perihelion distance of 0.331 AU and an eccentricity...

The orbit of the comet Encke has a perihelion distance of 0.331 AU and an eccentricity of 0.850. Suppose that the arrival of the comet at its perihelion distance happens to occur at a point directly in line between the Sun and the Earth. Roughly how many years will it be before such an event happens again?

The time period of an elliptical orbit is

Where M is the mass of sun = 1.989*1030 kg

a, the semi major axis is given by

a = q/(1-e)

q is the perihelion distance = 0.331 AU

e = 0.850

So, a = 0.331/(1-0.85) = 2.2067 AU = 2.2067*1.496*1011 m = 3.3012232*1011 m

G = 6.674*10-11 m3/kgs2

So,

The orbits will come in line, when the period of orbit of comet is an integral multiple of the period of earth.

That is: n*T becomes almost a whole number.

For n = 4, 4*3.27 = 13.1, which is close to a whole number. So, the event will happen after 13 years

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