Radiation Force

Consider an idealized Sun and Earth, both blackbodies, in otherwise empty flat space. The Sun is at a temperature T_S = 6000 K, and heat transfer by oceans and atmosphere on the Earth is so effective as to keep the Earth’s surface temperature uniform. The radius of the Earth is R_E = 6.4 × 10⁶ m, the radius of the Sun is R_S = 7 × 10⁸ m, and the Earth–Sun distance is d = 1.5 × 10¹¹m. The mass of Sun M_S = 2 × 10³⁰ kg.

a) Find the temperature of the Earth.

d) Find the radiation force on the Earth.

c) Compare these results with those for an interplanetary “chondrule” in the form of a spherical, perfectly conducting blackbody with a radius R = 0.1 cm, moving in a circular orbit around the Sun at a radius equal to the Earth–Sun distance d.

b) At what distance from the Sun would a metallic particle melt (melting temperature T_m = 1550 K)?

e) For what size particle would the radiation force calculated in (c) be equal to the gravitational force from the Sun at a distance?

SOLUTION

a) The total radiation flux from the Sun is

(1)

where σ is the Stefan–Boltzmann constant. Only a fraction π R²_E/4πd² of this flux reaches the Earth. In equilibrium this fraction equals the total flux radiated from the Earth at temperature T_E. So

(2)

From (2) we obtain

(3)

b) The radiation pressure on the Earth is given by

(4)

where (R_S/d)² is the ratio of the total flux from the Sun to the flux that reaches the Earth. The radiation force on the Earth

(5)

where A_E is the cross section of the Earth.

c) For the small “chondrule” the temperature will be the same because it depends only on the angle at which the Sun is seen and the radiation force:

(6)

d) Using (3) and denoting the melting temperature of the metallic particle T_m and the distance from the Sun d_c, we obtain

(7)

e) Let us estimate the radius of a particle for which the radiation force will equal the gravitational force at the distance of the Earth’s orbit d. Using (6), we have

(8)

where the particle mass m_p = (4/3)πr³ ρ, and ρ ~ 10³ kg/m³

(9)