Before You Begin

Question #42271

Difficulty: 8/10 | 🟢 Free

The Question

A planet of mass $M$ and cross-sectional area $A_p$ is in a nearly circular orbit of radius $R$ around a star of mass $M_s$ . The star emits a steady, spherically symmetric stellar wind of gas radially outwards at a constant speed $v_w$ . The total mass loss rate of the star due to this wind is $\dot{m}$ . The planet experiences a tangential drag force of magnitude $F_D = C_d \rho A_p v^2$ , where $\rho$ is the local gas density, $v$ is the planet's orbital speed, and $C_d$ is the drag coefficient. Assuming $M \ll M_s$ and $v_w \ll v$ , what is the magnitude of the rate of change of the planet's areal velocity, $\left| \frac{dV_A}{dt} \right|$ ?

Options

A.
$\frac{C_d \dot{m} A_p G M_s}{8\pi M R^2 v_w}$
B.
$\frac{C_d \dot{m} A_p G M_s}{4\pi M R^2 v_w}$
C.
$\frac{C_d \dot{m} A_p G M_s}{2\pi M R^2 v_w}$
D.
$\frac{C_d \dot{m} A_p G M_s}{16\pi M R^2 v_w}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Kepler’s Second Law (Areal Velocity)|Equation of Continuity|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of fluid continuity, orbital mechanics, and non-conservative forces.

A common trap is assuming Kepler's Second Law (constant areal velocity) holds universally, ignoring that drag provides a tangential torque.

Students may also incorrectly attempt to use energy conservation to find the orbital decay, which complicates the extraction of angular momentum loss.

Furthermore, the presence of the star's variable mass ( $\dot{m}$ ) acts as a conceptual decoy; while it causes the orbit to expand ( $M_s R = \text{const}$ ), central mass loss exerts no torque and thus does not affect the areal velocity.

The only source of areal velocity decay is the drag torque, requiring the correct application of $V_A = L/2M$ and the continuity equation $\rho = \dot{m}/(4\pi R^2 v_w)$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Kepler’s Second Law (Areal Velocity)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free