A planet of mass m moves around the sun of mass M in elliptical orbit. The maximum and minimum distances of the planet from the sun are r1 and r2 respectively.The time period of the planet is proportional to :

Question

A planet of mass m moves around the sun of mass M in elliptical orbit. The maximum and minimum distances of the planet from the sun are r1 and r2 respectively.The time period of the planet is proportional to : (A)  r12/5  (B) (r1+r2)3 / 2 (C) (r1-r2)3 / 2 (D)  r13/2

Tardigrade Admin · Accepted Answer

In astronomy the point: of closest approach is called the pericentre, ạnd that of farthest excursion is called apocentre: When

a

is semimajor axis we have
Periapsis

= (1 - e) a

.
Apoapsis

= (1 + e) a -

From condition given in question,

r_{1} = a (1 + e)

r_{2} = a (1 - e)

\Rightarrow r_{1} + r_{2} = 2 a

\Rightarrow a = \frac{r _{1} + r _{2}}{2}

Also from Kepler's third law of planetary motion we have, the square of the period of revolution

(T)

of any planet around the sun is directly proportional to the cube of the semimajor axis of its elliptic orbit

(a)

.
i.e.,

T^{2} \propto a^{3} 4

\Rightarrow T = a^{3/2}

\propto (\frac{r _{1} + r _{2}}{2})^{3/2}

\Rightarrow T \propto (r_{1} + r_{2})^{3/2}

Q. A planet of mass $m$ moves around the sun of mass $M$ in elliptical orbit. The maximum and minimum distances of the planet from the sun are $r_{1}$ and $r_{2}$ respectively.The time period of the planet is proportional to :

$r_{1}^{2/5}$

$(r_{1} + r_{2})^{3/2}$

$(r_{1} - r_{2})^{3/2}$

$r_{1}^{3/2}$

Q. A planet of mass m moves around the sun of mass M in elliptical orbit. The maximum and minimum distances of the planet from the sun are r1​ and r2​ respectively.The time period of the planet is proportional to :

r12/5​

(r1​+r2​)3/2

(r1​−r2​)3/2

r13/2​

Q. A planet of mass $m$ moves around the sun of mass $M$ in elliptical orbit. The maximum and minimum distances of the planet from the sun are $r_{1}$ and $r_{2}$ respectively.The time period of the planet is proportional to :

$r_{1}^{2/5}$

$(r_{1} + r_{2})^{3/2}$

$(r_{1} - r_{2})^{3/2}$

$r_{1}^{3/2}$