Question

Consider an encounter between two stars of mass $m$ where the nature of the encounter is typical of the sorts of encounters that normally occur in galaxies (i.e., a very weak deflection of one star's path). Star 1 is initially on a straight line trajectory and, if it were not deflected by star 2, the distance of closest approach between the two stars would be $b$ (this is known as the "impact parameter"). Let $\vec{v}$ be the relative velocity between the two stars when star 1 is at a distance $r >> b$. Suppose that star 2 causes only a very small deflection of star 1 from its original path, such that the relative velocity of star 1 is changed by an amount $\delta \vec{v}$, where $|\delta \vec{v}|/v << 1$. The goal of this problem is to estimate $|\delta v_\perp|$, the component of $\delta \vec{v}$ that is perpendicular to the original trajectory of star 1. The deflection of star 1 is due to the force of gravity, and the component of the force of gravity that causes $|\delta \vec{v}|$ to be non-zero is the component of the force that's perpendicular to the original trajectory of star 1: $F_\perp = m\vec{a}_\perp = m \frac{d\vec{v}_\perp}{dt}$. By finding an appropriate expression for $F_\perp$ we can perform an integral to obtain $|\delta v_\perp|$. a) From the diagram below, it's clear that $F_\perp = Gm^2r^{-2}\cos\theta$. Using the geometry of the diagram below, rewrite $F_\perp$ in terms of $b$ and $x$ (i.e., instead of in terms of $r$ and $\theta$). b) Since we're considering only a very small deflection, it's fair to adopt the approximation $x \approx vt$ because $|\vec{v}| = v$ is approximately constant. Use this to obtain an expression for $F_\perp$ as a function of time and, hence, an expression for $\frac{dv_\perp}{dt}$ as a function of time. c) Choose the time $t = 0$ to correspond to the time at which star 1 is located at the distance of closest approach. That is, let star 1 be at its starting location at $t = -\infty$ and at its ending location at $t = +\infty$. Now make the approximation $|\delta v_\perp| \approx dv_\perp$ and perform an appropriate integration to obtain an expression for $|\delta v_\perp|$. Express your final answer in terms of $m$, $b$, and $v$. You MAY NOT use a table of integrals or an on-line integrator to solve this problem.

Name: consider an encounter between two stars of mass m where the nature of the encounter is typical of the sorts of encounters that normally occur in galaxies ie a very weak deflection of one sta 75757
Uploaded: 2021-09-14T03:12:03-08:00
Duration: 4 min 26 s
Channel: Khoobchandra Agrawal
Description: consider an encounter between two stars of mass m where the nature of the encounter is typical of the sorts of encounters that normally occur in galaxies ie a very weak deflection of one sta 75757

          Consider an encounter between two stars of mass $m$ where the nature of the encounter is typical of the sorts of encounters that normally occur in galaxies (i.e., a very weak deflection of one star's path). Star 1 is initially on a straight line trajectory and, if it were not deflected by star 2, the distance of closest approach between the two stars would be $b$ (this is known as the "impact parameter").

Let $\vec{v}$ be the relative velocity between the two stars when star 1 is at a distance $r >> b$. Suppose that star 2 causes only a very small deflection of star 1 from its original path, such that the relative velocity of star 1 is changed by an amount $\delta \vec{v}$, where $|\delta \vec{v}|/v << 1$. The goal of this problem is to estimate $|\delta v_\perp|$, the component of $\delta \vec{v}$ that is perpendicular to the original trajectory of star 1.

The deflection of star 1 is due to the force of gravity, and the component of the force of gravity that causes $|\delta \vec{v}|$ to be non-zero is the component of the force that's perpendicular to the original trajectory of star 1: $F_\perp = m\vec{a}_\perp = m \frac{d\vec{v}_\perp}{dt}$. By finding an appropriate expression for $F_\perp$ we can perform an integral to obtain $|\delta v_\perp|$.

a) From the diagram below, it's clear that $F_\perp = Gm^2r^{-2}\cos\theta$. Using the geometry of the diagram below, rewrite $F_\perp$ in terms of $b$ and $x$ (i.e., instead of in terms of $r$ and $\theta$).

b) Since we're considering only a very small deflection, it's fair to adopt the approximation $x \approx vt$ because $|\vec{v}| = v$ is approximately constant. Use this to obtain an expression for $F_\perp$ as a function of time and, hence, an expression for $\frac{dv_\perp}{dt}$ as a function of time.

c) Choose the time $t = 0$ to correspond to the time at which star 1 is located at the distance of closest approach. That is, let star 1 be at its starting location at $t = -\infty$ and at its ending location at $t = +\infty$. Now make the approximation $|\delta v_\perp| \approx dv_\perp$ and perform an appropriate integration to obtain an expression for $|\delta v_\perp|$. Express your final answer in terms of $m$, $b$, and $v$. You MAY NOT use a table of integrals or an on-line integrator to solve this problem.

consider an encounter between two stars of mass m where the nature of the encounter is typical of the sorts of encounters that normally occur in galaxies ie a very weak deflection of one sta 75757

Added by Esther M.

Question

Please give Ace some feedback