Poynting-Robertson Effect

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<latex> \documentclass[11pt]{article} \def\inv#1Template:1 \over \def\ddtTemplate:D \over dt \def\mean#1{\left\langle #1\right\rangle} \def\sigot{\sigma_{12}} \def\sigto{\sigma_{21}} \def\eval#1{\big|_{#1}} \def\tr{\nabla} \def\dce{\vec\tr\times\vec E} \def\dcb{\vec\tr\times\vec B} \def\wz{\omega_0} \def\ef{\vec E} \def\ato{{A_{21}}} \def\bto{{B_{21}}} \def\bot{{B_{12}}} \def\bfieldTemplate:\vec B \def\apTemplate:A^\prime \def\xp{{x^{\prime}}} \def\yp{{y^{\prime}}} \def\zp{{z^{\prime}}} \def\tp{{t^{\prime}}} \def\upxTemplate:U x^\prime \def\upyTemplate:U y^\prime \def\e#1{\cdot10^{#1}} \def\eikrwt{e^{i(\vec k\vec r-wt)}}

\usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \usepackage{graphicx} \begin{document} \def\lya{Ly\alpha}

\section{What is Poynting-Robertson Drag?}


Dust grains orbiting a central star are affected by both the gravitational force, as well as photons emitted from the star.

The gravitational force is given simply by:

\begin{equation}

\vec{F}_g = -\frac{GMm}{r^2}\hat{r}    

\end{equation}

\noindent where $G$ is the gravitational constant, $M$ is the mass of the star, $m$ is the dust grain's mass, $r$ is the distance from the dust grain to the star, and $\hat{r}$ is the radial direction from the star. This force alone would cause the dust to orbit the star on a circular, Keplerian orbit.

Photons emitted from the star will be absorbed by the dust grain and then re-emitted. While the re-emission of energy will be isotropic in the grain's frame of reference (assuming the grain is spherical and made of the a homogeneous material), it will not be isotropic in the Sun's frame due to the relative velocity of the grain compared to the star. This will give rise to a non-radial 'drag' force, called Poynting-Robertson drag, which will result in the slow spiral of the grain in towards the star, where it will eventually be destroyed by dust sublimation or sputtering, or collide to form much smaller grains which are then launched out of the solar system on hyperbolic orbits (since, for smaller grains, the radiation pressure will be much larger than the graviational force).

\section{P-R Drag Derivation}

Note that this derivation is non-relativistic, and so is not strictly accurate, but gives a good picture of how and why P-R drag arises. For a more accurate (and complex) derivation and discussion see, e.g. Klacka 1992.

First, consider the radiation pressure force on the dust grain. Let $S$ be the energy flux [erg cm$^{-2}$ s$^{-1}$] at the location of a stationary dust grain a distance $r$ away from the star. If the particle has any component of its velocity in the direction from which the photons are streaming, then the flux of photons impacting the grain must be adjusted due to the Doppler shift as:

\begin{equation}

   S' = S\left(1 - \frac{\vec{v}\cdot\hat{S}}{c}\right)

\end{equation}

\noindent where $\vec{v}$ is the grain velocity as measured in the frame of the star and $\hat{S}$ is the direction from which the photons are streaming, also in the frame of the star.

The absorbed energy flux is therefore $S'\sigma_\textrm{dust}$, where $\sigma_\textrm{dust}$ is the absorption cross section of the dust grain to the incoming photons, which can be reasonably taken to be the grain's geometrical cross-section. The force due to radiation pressure is therefore:

\begin{equation}

   \vec{F}_r = \frac{S'\sigma_\textrm{dust}}{c}\hat{S}

\end{equation}

However, we still need to consider the effect of the particle re-radiating the energy. As mentioned, we assume that this re-radiation will be isotropic in the grain's frame, meaning it will be non-isotropic in the star's frame. Via $E=mc^2$, the energy radiated in the grain's frame is equivalent to a mass loss rate of $S'\sigma_\textrm{dust}/c^2$, which, in the star's frame, gives a momentum flux of $-(S'\sigma_\textrm{dust}/c^2)\vec{v}$, which effectively creates a drag on the particle. Thus, the total force on a perfectly absorbing particle due to radiation is:

\begin{equation}

   \vec{F} = \frac{S'\sigma_\textrm{dust}}{c}\hat{S} - \frac{S'\sigma_\textrm{dust}}{c^2}\vec{v}

\end{equation}

Rewriting this and keeping only factors to order $v/c$ (since the dust velocity will be much less than the speed of light), we have:

\begin{equation}

   \vec{F} = \frac{S\sigma_\textrm{dust}}{c}\left(\left(1 - \frac{\vec{v}\cdot\hat{S}}{c}\right)\hat{S} - \frac{\vec{v}}{c}\right)

\end{equation}

\section{Sample Model: Dust affected by PR drag}

Simplifying to a 2D system, and using the solar luminosity and mass, we can plot the distance of a grain of dust to the Sun as a function of time. For a 10$\mu$m dust grain with density 2.5g/cm$^3$, the distance as a function of time is plotted in the figure below. Notice that the dust moves from close to the radius of Earth to within the radius of mercury on the timescale of several thousand years. For larger dust grains, the timescale can increase up to hundreds of thousands of years.

\begin{figure*}[ht] \centering \includegraphics[scale = .35]{PR_drag.png} \caption{Distance from the sun for a dust grain acted on by PR drag as a function of time.} \label{fig:TID_lag} \end{figure*}[[File:File:Example.jpg]]

\end{document} <\latex>