A planet is revolving round the sun in elliptical orbit....

A planet is revolving round the sun in elliptical orbit. Velocity at perigee position (nearest) is $\mathrm{v}_{1} \mid$ and at apogee position (farthest) is $\mathrm{v}_{2}$ Both these velocities are perpendicular to the joining center of sun and planet $r$ is the minimum distance and $\mathrm{r}_{2}$ the maximum distance. (1) when the planet is at perigee position, it wants to revolve in a circular orbit by itself. For this value of $\mathrm{G}$ (A) Should increase (B) Should decrease (C) data is insufficient (D) will not depend on the value of $\mathrm{G}$ (2) At apogee position suppose speed of planer is slightly decreased from $\mathrm{v}_{2}$, then what will happen to minimum distance $r_{1}$ in the subsequent motion (A) $r_{1}$ and $r_{2}$ both will decreases (B) $\mathrm{r}_{1}$ and $\mathrm{r}_{2}$ both will increases (C) $\mathrm{r}_{2}$ will remain as it is while $\mathrm{r}_{1}$ will increase (D) $\mathrm{r}_{2}$ will remain as it is while $\mathrm{r}_{1}$ will decrease

A planet is revolving round the sun in elliptical orbit. Velocity at perigee position (nearest) is $\mathrm{v}_{1} \mid$ and at apogee position (farthest) is $\mathrm{v}_{2}$ Both these velocities are perpendicular to the joining center of sun and planet $r$ is the minimum distance and $\mathrm{r}_{2}$ the maximum distance.
(1) when the planet is at perigee position, it wants to revolve in a circular orbit by itself. For this value of $\mathrm{G}$
(A) Should increase
(B) Should decrease
(C) data is insufficient
(D) will not depend on the value of $\mathrm{G}$
(2) At apogee position suppose speed of planer is slightly decreased from $\mathrm{v}_{2}$, then what will happen to minimum distance $r_{1}$ in the subsequent motion
(A) $r_{1}$ and $r_{2}$ both will decreases
(B) $\mathrm{r}_{1}$ and $\mathrm{r}_{2}$ both will increases
(C) $\mathrm{r}_{2}$ will remain as it is while $\mathrm{r}_{1}$ will increase
(D) $\mathrm{r}_{2}$ will remain as it is while $\mathrm{r}_{1}$ will decrease

Key Concepts

Elliptical Orbits and Orbital Parameters

Elliptical orbits are characterized by two key points: the closest approach (pericenter) and the farthest point (apocenter) from the central body. The variations in speed along the orbit, with maximum speed at pericenter and minimum at apocenter, are inherent to elliptical orbits. This concept is fundamental to orbital mechanics as it highlights how the orbit’s shape and size are determined by the energy and angular momentum of the orbiting body.

Conservation of Energy in Orbital Mechanics

In orbital dynamics, the total mechanical energy—which is a sum of kinetic and potential energy—is conserved in the absence of external forces. Changes in velocity at specific points in the orbit, such as a small decrease in speed, alter the total energy and hence the shape of the orbit. This conservation principle helps predict how modifications in speed affect orbital parameters like the distances at pericenter and apocenter.

Gravitational Constant (G) in Orbital Dynamics

The gravitational constant is a universal constant that appears in Newton’s law of gravitation, defining the strength of the gravitational force between two masses. In orbital mechanics, G is a fixed value and is not subject to change; rather, it sets the scale for the gravitational attraction that governs the orbital motion of planets and satellites. This concept is central as it underpins the equations that relate orbital velocity, radius, and the central mass, irrespective of the shape of the orbit.

Circular Orbit Conditions and Centripetal Force

A circular orbit is maintained when the gravitational force provides exactly the centripetal force required to keep an object moving in a circle. In this scenario, the orbital velocity is determined by the balance between gravitational pull and the inertia of the moving object. Understanding this balance is essential for comparing circular and elliptical orbits, especially when considering how changes in velocity at a given radius imply a departure from circular motion.

Conservation of Angular Momentum

Angular momentum is conserved in orbital motion when no external torques are acting on the system. This conservation law ensures that the product of orbital radius and the tangential component of velocity remains constant throughout the orbit. It plays a critical role in relating the velocities at different orbital points and in understanding how impulsive changes in speed at certain points (such as at apocenter) can alter the orbit’s eccentricity and the distances of closest and farthest approach.

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