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Gravitational Waves from Binary Orbits
Developed by Robert Hilborn  Published September 10, 2017
This set of exercises leads the students through a visual simulation of binary orbit inspirals and the resulting gravitational wave signals observed by the LIGOVIRGO collaboration. The student should be able to understand the connection between the orbit inspiral and the gravitational waveform. In addition, the student should understand how the masses of the orbiting objects can be estimated from the gravitational waveform and how the waveform amplitude can be used to estimate the distance between the binary system and the observatory on Earth.
Subject Areas  Mechanics and Astronomy/Astrophysics 

Levels  First Year and Beyond the First Year 
Available Implementations  Glowscript and Mathematica 
Learning Objectives 
* Students will be able to download gravitational wave signal data from the LIGO Open Science site and to plot the gravitational wave signals. Students will be able to describe the general features of the observed waveform. (Exercise 1)
* Students will be able to use a visualization of the binary orbit inspiral and a graph of the corresponding gravitational waveform to understand how the waveform evolves in time and how the waveform is related to the LIGO data in Exercise 1. (Exercise 2)
* Students will be able to explain the connection between the orbital properties and the gravitational waveform as they evolve in time. (Exercise 3)
* Students will be able to determine a waveform frequency and an estimate of the rate of change of that frequency from a plot of that waveform. With that information, students will be able to estimate the binary masses and their separation from the gravitational waveform data. (Exercise 4)
* Students will be able to estimate the distance from the binary system to Earth using the amplitude of the gravitational waveform. (Exercise 5)
* Students will be able to predict the gravitational waveform for two orbiting neutron stars and to explain how the orbits scale with the masses of the orbiting objects. Students will be able to modify animation code in GlowScript or Mathematica to illustrate binary orbits for the binary neutron stars. (Exercise 6)

Time to Complete  120 min 
These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).
###Exercise 1
Go to the [Ligo Open Science Site](https://www.gwopenscience.org/events/GW150914/) for the GW strain data for the event of September 14, 2015: GW150914. Scroll down to find Fig. 1. In the second row of panels in Fig. 1, click on "click for DATA (Numerical Relativity)". Copy and paste those data into a spreadsheet.
Note: You may need to use the Excel feature Data > Text to Columns to separate the time data from the strain data. In the original LIGOVirgo data files, those data are separated by a space.
Those data are the results of using numerical general relativity calculations to predict the gravitational waveform data observed, taking into account the limited frequency bandwidth of the LIGO detector. The data are time (in seconds) and the strain signal $h(t)$. Plot those data.
Do the same for the data in Fig. 2 (left panel) (also labeled Numerical relativity Data). This data set is the prediction of numerical general relativity without the frequency limitations of the detector.
Your plots should be the same as those shown in the appropriate panels in LIGO's Fig. 1 and Fig. 2.
For both of those data sets, describe the general characteristics of the observed waveform. For example, how do the frequency and amplitude of the waveform change with time?
###Exercise 2
Use the program at [Binary Inspiral](http://www.glowscript.org/#/user/rhilborn/folder/Public/program/BinaryInSpiral) (this program should run in any browser) to see a visual simulation of the decay of binary orbits as the system emits gravitational wave energy. The program also plots the gravitational waveform that would be observed by the LIGOVIRGO interferometer observatory.
Describe the general features of the gravitational waveform, in particular how its frequency and amplitude evolve in time.
Note that the simulation does not include what happens in the last few milliseconds of the actual signal where there is a sudden jump in both amplitude and frequency. Those features are attributed to the merging and "ring down" of the two black holes as they come together to form a single black hole. The simulation is based on a linear version of general relativity and does not contain any information about the objects and what happens to them when they "collide."
How are those changes in amplitude and frequency related to what is happening to the with the binary orbit?
It turns out that the gravitational wave frequency is twice the orbital frequency. If you watch the simulation carefully, you can tap your foot once per orbital cycle and see that there are two cycles of the GW waveform during that time period.
###Exercise 3
Review the theory section to see how the energy radiated by the binary system depends on the orbital frequency and the mass separation. View the simulation again and explain how the plotted waveform properties are in general agreement with the theory. Hint: It is helpful to use Kepler's Third Law to put the results entirely in terms of the orbital frequency or entirely in terms of the orbital separation.
For the LIGOVIRGO GW150914 detection event, the detailed analysis showed that the masses were about 36 and 29 solar masses and the separation was about 10^6 meters. From that information calculate the orbital frequency and compare that to the observed wave frequency. Hints: recall that the wave frequency is twice the orbital frequency. Recall the difference between angular frequency (radians/s) and ordinary frequency (Hz).
Why did the LIGOVIRGO collaboration conclude that the objects must be black holes? Hints: Compare the 10^6 m separation to the radius of the Sun. Compare the spatial separation between the two objects $r(t)$ with the radius of a typical 30 solarmass star (How should the radius of a star scale with its mass, at least approximately?) and with the radius of a black hole (the Schwartzschild radius) with a mass of about 30 solar masses. Recall that the Schwartzschild radius of a spherical object with mass $M$ is given by
$$r_S = 2GM/c^2$$
###Exercise 4
Use the [Binary Inspiral](http://www.glowscript.org/#/user/rhilborn/folder/Public/program/BinaryInSpiral) program to estimate the total mass of the binary system used in the simulation using the following procedures. Once the graph of $h(t)$ is produced, your cursor can be used to read off coordinates from the graph. According to the theory section you need to known $\dot\omega$ and $\omega$ to find $\eta^{3/5} M$. Decide how to find $\dot\omega$ and $\omega$ from the graph and then compare your results to the mass values used in the simulation. Notice that $\eta \simeq 0.25$ if the masses are approximately equal.
Once you have found a value for the mass, calculate the mass separation that leads to the observed frequency. (Again remember that the wave frequency is twice the orbital frequency and recall the difference between angular frequency (radians/s) and wave frequency (Hz).)
###Exercise 5
The theory section states that the amplitude of the strain signal is related to the mass separation and the Schwartzschild radius of the total mass of the binary system:
$$h = \frac{\eta}{8} \frac{r_S^2}{r R}$$
ignoring angular factors dealing with polarization, inclination of the binary orbits etc. Use $h \simeq 10^{21}$ (the LIGOVIRGO result) to estimate the distance from the binary system to Earth.
From that result, is the binary system relatively close by (say within the Milky Way Galaxy) or far away? Compare that distance to the size of the known universe.
###Exercise 6
The LIGOVIRGO collaborative announced in late 2017 the detection of gravitational waves from two orbiting neutron stars (Phys. Rev. Lett. 119, 161101 (2017)). Look up information on the typical mass of a neutron star. Use the Code provided in this Exercise Set to plot the expected waveform. Please remember that to edit the GlowScript cord, you need to copy the code to an empty GlowScript file. You will need to find a reasonable range of initial and final separations for the plot. Explain why the binary neutron star orbit spirals in more slowly than the binary black hole orbit.
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Credits and Licensing
Robert Hilborn, "Gravitational Waves from Binary Orbits," Published in the PICUP Collection, September 2017.
The instructor materials are ©2017 Robert Hilborn.
The exercises are released under a Creative Commons AttributionNonCommercialShareAlike 4.0 license