Physics

Physics · mechanics

Problem

A.2 The total energy of the 2-mass system, in circular orbits, can be expressed as: E=A(μ,Ω,L)−GMμL,(3)E=A(\mu, \Omega, L)-G \frac{M \mu}{L}, \tag{3} where μ≡M1M2M1+M2,M≡M1+M2(4)\mu \equiv \frac{M_{1} M_{2}}{M_{1}+M_{2}}, \quad M \equiv M_{1}+M_{2} \tag{4} are the reduced mass and total mass of the system, Ω\Omega is the angular velocity of each mass and LL is the total separation L=r1+r2L=r_{1}+r_{2}. Obtain the explicit form of the term A(μ,Ω,L)A(\mu, \Omega, L). Given constants: LIGO-GW150914 In 2015, the gravitational-wave observatory LIGO detected, for the first time, the passing of gravitational waves (GW) through Earth. This event, named GW150914, was triggered by waves produced by two black holes that were orbiting on quasi-circular orbits. This problem will make you estimate some physical parameters of the system, from the properties of the detected signal. Part A: Newtonian (conservative) orbits Context question: A.1 Consider a system of two stars with masses M1,M2M_{1}, M_{2}, at locations r⃗1,r⃗2\vec{r}_{1}, \vec{r}_{2}, respectively, with respect to the center-of-mass of the system, that is, M1r1→+M2r2→=0.(!)M_{1} \overrightarrow{r_{1}}+M_{2} \overrightarrow{r_{2}}=0 . \tag{!} The stars are isolated from the rest of the Universe and moving at nonrelativistic velocities. Using Newton's laws, the acceleration vector of mass M1M_{1} can be expressed as d2r⃗1 dt2=−αr⃗1r1n(2)\frac{\mathrm{d}^{2} \vec{r}_{1}}{\mathrm{~d} t^{2}}=-\alpha \frac{\vec{r}_{1}}{r_{1}^{n}} \tag{2} where r1=∣r⃗1∣,r2=∣r⃗2∣r_{1}=\left|\vec{r}_{1}\right|, r_{2}=\left|\vec{r}_{2}\right|. Find n∈Nn \in \mathbb{N} and α=α(G,M1,M2)\alpha=\alpha\left(G, M_{1}, M_{2}\right), where GG is Newton's constant [G≃6.67×10−11 N m2 kg−2]\left[G \simeq 6.67 \times 10^{-11} \mathrm{~N} \mathrm{~m}^{2} \mathrm{~kg}^{-2}\right]. Context answer: n=3n=3 , α=GM23(M1+M2)2\alpha=\frac{G M_{2}^{3}}{(M_{1}+M_{2})^{2}}

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