- #1
leo_africanus
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I'm in the 11th hour of studying, and have run into a problem I can't seem to tackle from an old exam. I may be making some foolish mistakes, as I've now been staring at this thing for entirely too long.
1. Homework Statement
(a) Calculate the force due to radiation pressure experienced by an object of radius, r, and density, ρ, in a circular orbit with semimajor axis, a, around the Sun. Assume that the object absorbs all radiation and re-emits it isotropically in its rest-frame.
(b) If the object were stationary, this force would act only in the radial direction away from the Sun. However, because of our object’s orbital velocity, the direction of the incoming photons has a small non-radial component in the object’s rest-frame, and the radiation pressure exerted by the Sun in part a) has a small non-radial component. Expressing the object’s orbital velocity in terms of the Sun’s mass and a, solve for the non-radial component of the radiation pressure. (This non-radial component can be thought of as a photon headwind known as Poynting-Robertson drag.) [As a side note- this explanation isn't quite correct, right? Or is this just a different way or thinking about it? I'd always understood that the force in P-R effect came from the object re-emitting the absorbed radiation as it moves, making it anisotropic from to stationary reference frame]
(c) This headwind causes the orbital semimajor axis to decay over time. Write the time derivative of the semimajor axis due to Poynting-Robertson drag for the object in part a. Assume that the radial component of the radiation pressure force is very small compared to the Sun’s gravitational pull, so we only need to consider the Poynting-Robertson component.
No equations given, but I've included the ones I used as they come up in my attempt at a solution below (hope this is okay).
a) The cross-sectional area of the object is ## A=\pi r^2 ##. If we divide this by the surface area of a sphere with radius a, ##S=4\pi a^2##, we get the fraction of the total radiation pressure from the sun that will impart a force on the object: ##\frac{r^2}{4a^2}##
We then need to calculate the total force from the sun's photons, which is simply ##\frac{L_{\odot}}{c}##, and multiply by our fraction from before, giving us: ##F_{rad}=\frac{r^2}{4a^2} \frac{L_{\odot}}{c}## (where we can of course plug in values for c and ##L_{\odot}## to get an answer just in terms of a and r if we want)
b) I believe the force in the tangential direction is simply the ratio of the velocity in that direction and the velocity of the photons multiplied by the total force:
$$v_{orb}=\sqrt[]{\frac{GM_{\odot}}{a}}$$
$$F_{pr}=\frac{v_{orb}}{c} F_{rad} =\frac{r^2}{4a^2} \frac{L_{\odot}}{c^2} \sqrt[]{\frac{GM_{\odot}}{a}} = \frac{r^2}{4} \frac{L_{\odot}}{c^2} \sqrt[]{\frac{GM_{\odot}}{a^5}}$$
I feel pretty good about this answer so far, but I may have made a foolish mistake somewhere
c) This is where things get a bit fuzzier for me. I've tried two big approaches which both leave me stuck later on. I think I've just played with it enough that I've gotten myself lost and confused, so any help here is appreciated.
Firstly, I tried using ## F_{pr} = \frac{dp}{dt} ##, which after integrating gives me:
##p = \frac{r^2}{4} \frac{L_{\odot}}{c^2} \sqrt[]{\frac{GM_{\odot}}{a^5}} t + p_0## (where I assumed that the constant is the initial momentum at time t=0, ##p_0##)
##a^5 (p - p_0)^2 = \frac{r^4}{16} \frac{L_{\odot}^2}{c^4} G M_{\odot} t^2 ## (moving the ##p_0## over, squaring both sides, and moving the ##a^5## over)
This seemed fairly straight forward to solve for a and then take a time derivative until I realized that I needed to plug something in for p, which of course ends up being a function of a as well:
##p=m_{obj} v_{orb} = \frac{4}{3} \pi \rho r^3 \sqrt[]{\frac{GM_{\odot}}{a}}##
Leaving the mass of the object as m for now, this gives:##a^5 (m^2\frac{GM_{\odot}}{a} - 2 m \sqrt[]{\frac{GM_{\odot}}{a}} p_0 + p_0^2) = \frac{r^4}{16} \frac{L_{\odot}^2}{c^4} G M_{\odot} t^2 ##
Simplifying this a bit, I tried using quadratic formula to get a solution for a which I could then differentiate with respect to time, but I wasn't able to simplify anything out of the formula and so kind of assumed I was on the wrong track there.
The other approach I tried was using ##F= m_{obj} a_{orb} = m_{obj} \frac{dv_{orb}}{dt} ##. This ends with a similar un-useful (as far as I can tell) quadratic solution since ##v_{orb}## again depends on a, the semi-major axis.
Where have I gone wrong? Is the proper solution through relating force and the change in potential energy instead?
Any feedback is appreciated (be gentle if I've made very foolish errors-- I've been physics-ing too many hours this week).
1. Homework Statement
(a) Calculate the force due to radiation pressure experienced by an object of radius, r, and density, ρ, in a circular orbit with semimajor axis, a, around the Sun. Assume that the object absorbs all radiation and re-emits it isotropically in its rest-frame.
(b) If the object were stationary, this force would act only in the radial direction away from the Sun. However, because of our object’s orbital velocity, the direction of the incoming photons has a small non-radial component in the object’s rest-frame, and the radiation pressure exerted by the Sun in part a) has a small non-radial component. Expressing the object’s orbital velocity in terms of the Sun’s mass and a, solve for the non-radial component of the radiation pressure. (This non-radial component can be thought of as a photon headwind known as Poynting-Robertson drag.) [As a side note- this explanation isn't quite correct, right? Or is this just a different way or thinking about it? I'd always understood that the force in P-R effect came from the object re-emitting the absorbed radiation as it moves, making it anisotropic from to stationary reference frame]
(c) This headwind causes the orbital semimajor axis to decay over time. Write the time derivative of the semimajor axis due to Poynting-Robertson drag for the object in part a. Assume that the radial component of the radiation pressure force is very small compared to the Sun’s gravitational pull, so we only need to consider the Poynting-Robertson component.
Homework Equations
No equations given, but I've included the ones I used as they come up in my attempt at a solution below (hope this is okay).
The Attempt at a Solution
a) The cross-sectional area of the object is ## A=\pi r^2 ##. If we divide this by the surface area of a sphere with radius a, ##S=4\pi a^2##, we get the fraction of the total radiation pressure from the sun that will impart a force on the object: ##\frac{r^2}{4a^2}##
We then need to calculate the total force from the sun's photons, which is simply ##\frac{L_{\odot}}{c}##, and multiply by our fraction from before, giving us: ##F_{rad}=\frac{r^2}{4a^2} \frac{L_{\odot}}{c}## (where we can of course plug in values for c and ##L_{\odot}## to get an answer just in terms of a and r if we want)
b) I believe the force in the tangential direction is simply the ratio of the velocity in that direction and the velocity of the photons multiplied by the total force:
$$v_{orb}=\sqrt[]{\frac{GM_{\odot}}{a}}$$
$$F_{pr}=\frac{v_{orb}}{c} F_{rad} =\frac{r^2}{4a^2} \frac{L_{\odot}}{c^2} \sqrt[]{\frac{GM_{\odot}}{a}} = \frac{r^2}{4} \frac{L_{\odot}}{c^2} \sqrt[]{\frac{GM_{\odot}}{a^5}}$$
I feel pretty good about this answer so far, but I may have made a foolish mistake somewhere
c) This is where things get a bit fuzzier for me. I've tried two big approaches which both leave me stuck later on. I think I've just played with it enough that I've gotten myself lost and confused, so any help here is appreciated.
Firstly, I tried using ## F_{pr} = \frac{dp}{dt} ##, which after integrating gives me:
##p = \frac{r^2}{4} \frac{L_{\odot}}{c^2} \sqrt[]{\frac{GM_{\odot}}{a^5}} t + p_0## (where I assumed that the constant is the initial momentum at time t=0, ##p_0##)
##a^5 (p - p_0)^2 = \frac{r^4}{16} \frac{L_{\odot}^2}{c^4} G M_{\odot} t^2 ## (moving the ##p_0## over, squaring both sides, and moving the ##a^5## over)
This seemed fairly straight forward to solve for a and then take a time derivative until I realized that I needed to plug something in for p, which of course ends up being a function of a as well:
##p=m_{obj} v_{orb} = \frac{4}{3} \pi \rho r^3 \sqrt[]{\frac{GM_{\odot}}{a}}##
Leaving the mass of the object as m for now, this gives:##a^5 (m^2\frac{GM_{\odot}}{a} - 2 m \sqrt[]{\frac{GM_{\odot}}{a}} p_0 + p_0^2) = \frac{r^4}{16} \frac{L_{\odot}^2}{c^4} G M_{\odot} t^2 ##
Simplifying this a bit, I tried using quadratic formula to get a solution for a which I could then differentiate with respect to time, but I wasn't able to simplify anything out of the formula and so kind of assumed I was on the wrong track there.
The other approach I tried was using ##F= m_{obj} a_{orb} = m_{obj} \frac{dv_{orb}}{dt} ##. This ends with a similar un-useful (as far as I can tell) quadratic solution since ##v_{orb}## again depends on a, the semi-major axis.
Where have I gone wrong? Is the proper solution through relating force and the change in potential energy instead?
Any feedback is appreciated (be gentle if I've made very foolish errors-- I've been physics-ing too many hours this week).
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