Question

Members of the astronomy club just posted an article about the discovery of a small comet which orbit is in the same orbit plane as the Earth. In the article it is reported that the observation took place when the comet was eclipsed by the sun just passing the aphelion, in a relative distance at that moment of 3.82 UA and an orbit with eccentricity of e=0.777.

They are calling you to form part of the group of immediate reaction to determine if the comet is a rick to planet Earth. Do the calculation. Use Kepler laws to determine if the ellipse of the comet (period, formula, intersection points with the ellipse of the earth).

Answer 1

First I have calculated the necessary stuff for the comet, such as a, b, c and e. Then I have written the equation of the comet. For the earth, I have done the same and tried to show the orbits of both the objects diagrammatically for better understanding. In the diagram, we can see that if sun is the focus and lies at about origin, earth's orbit around it is almost circular. Then, making use of the fact that aphelion of comet is around 3.82 AU from sun, we can sketch the comet's path. On solving mathematically, we can find the intersection points. The quadratic solutions can be easily solved by discriminant method. We get a complex root, which is discarded, and a real root for each symmetricity of the comet's orbit.