Question
A long, thin (i.e., needle-shaped) dust grain floats in a box filled with gas at a constant temperature $T$. On average, is the angular momentum vector nearly parallel to or perpendicular to the long axis of the grain? Explain.
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The dust grain is described as long and thin, resembling a needle. This shape suggests that it has a well-defined long axis. Show more…
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In an orbital motion, the angular momentum vector is: (a) along the radius vector (b) parallel to the linear momentum (c) in the orbital plane (d) perpendicular to the orbital plane
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When a mass is rotating in a plane about a fixed axis, its angular momentum is directed along (a) the radius. (b) the tangent to the orbit. (c) the line at an angle of $5^{\circ}$ to the plane of rotation. (d) the axis of rotation.
A dust grain orbiting the Sun (or in a planetary ring system) absorbs and then re-emits solar radiation. since the light is radiated from the Sun isotropically and re-emitted by the grain preferentially in the direction of motion, the particle is decelerated (it loses angular momentum) and spirals in toward the object it is orbiting. This process (known as the Poynting-Robertson effect is just a consequence of the headlight effect. (a) If a dust grain orbiting the Sun absorbs $100 \%$ of the energy that strikes it and all of the energy is then re-radiated so that thermal equilibrium is maintained, what is the luminosity of the grain? Assume that the particle's cross-sectional area is $\sigma_{g}$ and its distance from the Sun is $r$ (b) Show that the rate at which angular momentum is lost from a grain is given by $$\frac{d \mathcal{L}}{d t}=-\frac{\sigma_{g}}{4 \pi r^{2}} \frac{L_{\odot}}{m c^{2}} \mathcal{L}$$ where $m$ and $\mathcal{L}=m v r$ are the mass and angular momentum of the grain, respectively, and $L_{\odot}$ is the luminosity of the Sun. Hint: Think of radiated photons as carrying an effective mass away from the grain; the effective mass of a photon is just $m_{y}=E_{y} / c^{2}$
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