This Exercise Set has been submitted for peer review, but it has not yet been accepted for publication in the PICUP collection.

Binary Orbit Inspirals

Developed by Robert Hilborn

This set of exercises takes a student through a series of calculations and computations to understand that the gravitational wave energy emitted by orbiting binaries causes the orbits to decay, leading to the so-called in-spiral of the orbits. The exercises should be relatively easy for students in junior-level mechanics courses but they are also accessible to students who have taken just calculus-based introductory physics, assuming that they have studied circular motion, Newtonian gravitation, and Kepler's Laws.
Subject Areas Mechanics and Astronomy/Astrophysics
Levels First Year and Beyond the First Year
Learning Objectives
* Students will be able to explain what properties of orbiting binaries are important in determining the gravitational wave energy emitted by the orbiting objects. (Exercise 1) * Students will be able to explain how Kepler's Third Law links the orbital angular frequency of the binaries to the separation between the binaries. They will be able to generate and evaluate realistic numbers for the orbital frequencies and separations for astrophysical situations. (Exercise 2) * Students will be able to explain quantitatively how much energy is emitted by gravitational waves from orbiting binaries with a specified orbital frequency or specified separation. (Exercise 3) * Students will be able to combine the gravitational wave energy emitted by the binaries with the Newtonian mechanics of the orbits to find how the orbits decay (the separation decreases) over time leading to the in-spiral of the orbits. (Exercise 4) * Students will be able to use, modify or produce a visual simulation of the binary in-spiral and use that to provide evidence to justify the claims that the orbital speed increases as the mass separation decreases. (Exercise 5)
Time to Complete 120 min
###Exercise 1 Using your own words, explain to a fellow student who has not read the theory section what properties of orbiting binaries are important in producing gravitational waves. Explain why the LIGO-VIRGO collaborative focused on finding gravitational radiation from orbiting black holes whose masses were several times a solar mass or larger. ###Exercise 2 Set up a spreadsheet or a simple computer program to find the Schwarschild radius corresponding to the sum of the two masses, and given, the separation between the masses, the orbital frequency in both radians/sec and Hz (sec^-1) and the relative change in the mass separation $\dot{r}/r$. Such a spreadsheet (or computer program) will be very useful in finding initial conditions for plots of the binary inspiral and in the Gravitational Waves set of exercises. It will be helpful to put the masses in units of the Sun's mass (the solar mass) since those units are used in much of gravitational wave physics. Use the spreadsheet to find the separation between two 35 solar mass objects that is needed to have the orbital frequency be about 100 radians per second. This orbital frequency puts the gravitational wave frequency in the range of frequencies to which the LIGO-VIRGO observatories are sensitive. Compare that separation to the Scharzschild radius for the total mass (which is the sum of the Schwarzschild radii of the individual objects). ###Exercise 3 For the conditions specified at the end of Exercise 2, calculate the total gravitational wave power emitted by the binary system. Compare that power to the luminosity of the Sun. ###Exercise 4 Set up a computer program (or spreadsheet) to find $r(t)$ and $\omega(t)$ for the conditions used at the end of Exercise 2. Note that we have an analytic expression for $r(t)$. Hint: the in-spiral lasts for only a few tenths of a second once the separation has reached the value specified in the last part of Exercise 2. Include plots of $r(t)$ and $r(t) \omega(t)/2$ which is the speed of one of the two equal masses. For the $r(t)$, plot, $r(t) /r_S$ and for the speeds, plot the speed divided by _c_. Set up the program (or the spreadsheet) so that it is easy to enter new mass values and new initial separations. Note that in our analysis, there is nothing to prevent speeds from exceeding _c_ or $r$ becoming smaller than $r_S$. ###Exercise 5 (optional, but fun) The previous analysis provides enough information for you to produce a visual simulation of the binary in-spiral. An example of this kind of simulation in Glowscript is available at [Binary Inspiral]( You can copy the code from that program and modify it. A Mathematica version is included in the Complete Codes section of this Exercise Set. In our model, we will assume slowly changing circular orbits, so we don't need to use Newton's Laws. The key elements are knowing the coordinates of the two masses in terms of $r(t)$. First, recall that for a central force law such as Newtonian gravitation, the orbits will lie in a plane. So we need worry only about _x_ and _y_ coordinates. Going back to the theory section, you should recognize that >$$r_a(t) =(m_b/M) r(t)$$ with an analogous expression for $r_b(t)$. Thus for mass $m_a$ we have, using the center of mass as the coordinate origin, the $x$ and $y$ coordinates are expressed as >$$x_a (t)=r_a(t) cos \omega(t)$$ >$$y_a (t)=r_a(t) sin \omega(t)$$ What are the corresponding coordinates for mass $m_b$? Hint: the two masses are always on opposite sides of the center of mass. Note that in this model both the radii of the orbits and their frequencies change (as they must in accordance with Kepler's Third Law). Using that information, it is relatively easy to produce a visual simulation of the in-spiral. This Exercise Set provides VPython (Glowscript) and Mathematica versions of the simulation. The Mathematica version uses the "Animate" command. Run the Exercise 5 simulations and explain the links between what the binary objects are doing and the theory. If you use GlowScript, which runs in almost any browser, you can show your results to your friends and families using your smart phone or any computer. The code provided gives "bare bones" simulations. If you are comfortable with VPython (GlowScript) or Mathamtica, try writing code to improve the utility of the simulation in understanding the binary inspiral.

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Credits and Licensing

The instructor materials are ©2017 Robert Hilborn.

The exercises are released under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 license

Creative Commons Attribution-NonCommercial-ShareAlike 4.0 license