A perfectly absorbing spherical dust grain of radius a and density r is in a circular orbit of radius R around a star of mass M and luminosity L. Calculate the speed of its orbit in terms of these variables, and show that it is slower than an orbiting planet due to the effects of radiation pressure. [Note that a planet feels no significant force from radiation.]
To calculate the speed of the dust grain's orbit, we need to consider the forces acting on it. In this case, there are two main forces to consider: gravity and radiation pressure.
1. Gravity Force:
The gravitational force acting on the dust grain can be calculated using Newton's law of universal gravitation:
F_gravity = G * (m * M) / r^2
where G is the gravitational constant, m is the mass of the dust grain, M is the mass of the star, and r is the distance between the dust grain and the center of the star.
2. Radiation Pressure:
The radiation pressure exerted on the dust grain by the star can be calculated using the formula:
F_radiation = (π * a^2 * I) / c
where a is the radius of the dust grain, I is the intensity of the star's radiation, and c is the speed of light.
For simplicity, let's assume that the dust grain is in a circular orbit around the star. In a circular orbit, the gravitational force provides the necessary centripetal force to keep the dust grain in orbit. So we can equate the gravitational force to the centripetal force:
F_gravity = F_centripetal
G * (m * M) / r^2 = (m * v^2) / r
where v is the speed of the dust grain in its orbit.
Now, let's substitute the expression for the radiation pressure force into the equation for F_centripetal:
G * (m * M) / r^2 = (m * v^2) / r + (π * a^2 * I) / c
We can rearrange this equation to solve for v:
v^2 = (G * M / r) - (π * a^2 * I) / (c * m)
Thus, we have derived the equation for the speed of the dust grain's orbit in terms of the given variables.
To show that the dust grain's orbit is slower than an orbiting planet due to the effects of radiation pressure, we compare its speed to the speed of a planet in orbit around the same star.
For a planet, the only significant force acting on it is gravity, so the speed of a planet in orbit is given by:
v_planet^2 = G * (M + m) / r
where m is the mass of the planet.
Comparing the expressions for v^2 and v_planet^2, it is evident that the presence of the additional term (π * a^2 * I) / (c * m) in the equation for v^2 reduces the value of v, making it slower than v_planet. This additional term represents the radiation pressure force acting on the dust grain, which is absent for planets.
Hence, we have shown that the dust grain's orbit is slower than an orbiting planet due to the effects of radiation pressure.