Question

A comet is first seen at a distance of d astronomical units from the Sun and it is traveling with a speed of q times the Earth’s speed. Show that the orbit of the comet is hyperbolic, parabolic, or elliptic, depending on whether the quantity q 2d is greater than, equal to, or less than 2, respectively

Answer 1

To solve this problem it is necessary to take into account the concepts of Gravitational Force and Kinetic Energy.

The kinetic energy is given by the equation:

$F= \frac{mv^2}2$

La energía gravitacional por,

$F=\frac{GM_cm}{d}$

Where m is the mass, v is the velocity, G the gravitational constant $M_e$ the mass of the earth, m the mass of the sun and d the distance ..

The sum of the energies, we must be a total energy

$E= \frac{mv^2}2+\frac{GM_em}{d}$

By the type of orbit we know that

E> 0 is a hyperbolic orbit

E = 0 is a parabolic orbit

E <0 is a closed orbit.

In the case of hyperbolic orbit

E>0

$\frac{mq^2}{2}-\frac{GM_em}{d}>0\\\frac{qv^2_e}{2}>\frac{GM_em}{d}\\q^2d>2\frac{GM_e}{v^2_e}\\q^2d>2$

The case of the comet is a closed orbit, so,

E<0

$\frac{mv^2}2+\frac{GM_em}{d}$

For parabolic orbit

E=0

$\frac{mv^2_eq^2}{2}-\frac{GM_cm}{d}=0\\\frac{v^2_eq^2}{2}=\frac{GM_c}{d}\\q^2d=2\frac{GM_e}{v^2_e}\\q^2d=2$

For the sun and the earth

$\frac{m_ev_e^2}{r}=\frac{GM_em_e}{r^2}$

$v^2_e=\frac{GM_e}{r}\\\frac{GM_e}{v_e}=r$

where $R \approx 1AU$

$q^2d$  For elliptical orbit