Solar Radiation Pressure and Poynting-Robertson Effect · USAPhO 2018 #B3

Physics · mechanics

Problem

Radiation pressure from the sun is responsible for cleaning out the inner solar system of small particles. a. The force of radiation on a spherical particle of radius r is given by

F = PQ\pi r^2

where P is the radiation pressure and Q is a dimensionless quality factor that depends on the relative size of the particle r and the wavelength of light λ. Throughout this problem assume that the sun emits a single wavelength

\lambda_{max}

; unless told otherwise, leave your answers in terms of symbolic variables. i. Given that the total power radiated from the sun is given by

L_{\odot}

, find an expression for the radiation pressure a distance R from the sun. ii. Assuming that the particle has a density ρ, derive an expression for the ratio

F_{radiation}/F_{gravity}

in terms of

L_{\odot}

, mass of sun

M_{\odot}

, ρ, particle radius r, and quality factor Q. iii. The quality factor is given by one of the following • If

r \ll \lambda

Q \sim (r/\lambda)^2

• If

r \sim \lambda

Q \sim 1

. • If

r \gg \lambda

Q = 1

Considering the three possible particle sizes, which is most likely to be blown away by the solar radiation pressure? b. The Poynting-Robertson effect acts as another mechanism for cleaning out the solar system. i. Assume that a particle is in a circular orbit around the sun. Find the speed of the particle v in terms of

M_{\odot}

, distance from sun R, and any other fundamental constants. ii. Because the particle is moving, the radiation force is not directed directly away from the sun. Find the torque τ on the particle because of radiation pressure. You may assume that

v \ll c

. iii. Since

\tau = dL/dt

, the angular momentum L of the particle changes with time. As such, develop a differential equation to find dR/dt, the rate of change of the radial location of the particle. You may assume the orbit is always quasi circular. iv. Develop an expression for the time required to remove particles of size

r \approx 1cm

and density

\rho \approx 1000kg/m^3

originally in circular orbits at a distance

R = R_{earth}

, and use the numbers below to simplify your expression. Some useful constants include

M_{\odot} = 1.989 \times 10^{30}kg

L_{\odot} = 3.828 \times 10^{26}W

R_{earth} = 1.5 \times 10^{11}m

\lambda_{max} = 500nm

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