Ultra-Compact Binary Systems and the Resulting GravitationalWaves

Sam CareyTerm Project

Astronomy 15, Dartmouth College

May 31, 2016

Abstract

In the Milky Way Galaxy, there exist a great deal of ultra-compact binary star systems thatconsist of white dwarfs, neutron stars, and other small, dense stars. An ultra-compact binary isdefined as a binary system with a very short orbital period: generally, less than one hour. Whenthe system is sufficiently massive, and has a high enough frequency, it will emit gravitationalwaves large enough to detect. However, only a few of these systems produce waves strongenough for detection by the LISA mission. This paper will examine the techniques of observ-ing the binary systems, analyze the structure and properties of some of the strongest emittingsystems, display calculations of certain properties of the systems and the gravitational wavesthey emit, and explain the future potential to advance our knowledge of the universe based onthe study of these systems.

1 System Classification

There are a number of types of ultra-compact binary systems, differing based on the physicalmake up of their orbiting stars. Due to their high frequency, and thus larger amplitude gravitationalwaves (see section Data, below, for an explanation of gravitational wave amplitude), scientistschoose to examine the systems in which the two stars orbit each other in the smallest amountof time. These high-energy systems are generally assumed to produce waves large enough to bedetected by human technology. In the table in the Data section, there are data from four differenttypes of ultra-compact binary systems, each unique in composition:

1. AM Canum Venaticorum Stars, or AM CVn Stars, are a rare type of binary that exist asa white dwarf (WD) primary accreting star and a smaller, secondary donor star in tandem.These stars are classified as cataclysmic variable stars, because they increase dramaticallyin brightness before returning to a dimmer state (Kilic et. al., 2013). Specifically, when theprimary star accretes a critical amount of hydrogen from the secondary star, the density andtemperature of the hydrogen layer increase sufficiently to initiate runaway hydrogen burning,converting the outer layer of hydrogen into helium and releasing a large amount of energy –hence the star’s variable luminosity (Nelemans et. al, 2010).

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White dwarfs are the remnant of lower-mass stars that have gone through a supernova.They are extremely dense, with masses comparable to that of the sun, and volumes compa-rable to that of the Earth. Included in this study are the prototype of this class, AM CanumVenaticorum, which consists of a 0.71 M� WD and a 0.13 M� companion orbiting in a 17minute period. Also, HM Cancri (or HM Cnc) is made of a 0.55 M� WD with a 0.27 M�companion in a 5.4 minute period (Kilic et. al, 2013).

2. X-Ray Binary Systems consist of an extremely dense star, called the accretor, and anothernormal star called the donor. The high mass accretor steals mass away from the lower massdonor as the donor orbits the accretor. The accretor can be a neutron star, ie, the smallest andmost dense type of star known in the universe. Neutron stars can have radii as small as 11kilometers, and masses of up to twice that of the sun (Kilic et. al, 2013). They result fromthe gravitational collapse of a heavy star, following that star’s termination of fusion and theresulting supernova. This type of system is named for the luminous X-rays that are emittedas a result of the accretion process. The systems in this study are 4U 1820-30, where a 1.4M� NS and a 0.06 M� WD orbit in a 685 second period; 4U 01513-40, where a 1.4 M� NSand a 0.05 M� WD orbit in 17 minutes; and RX J0806.3+1527, where two 0.5 M� WDsorbit in 5.3 minutes (Kilic et. al, 2013).

3. Double Pulsar Systems have two extremely dense, pulsing neutron stars in orbit aroundeach other. This study looks at the Hulse-Taylor Binary Pulsar, PSR B1913+16, in which1.44 M� NS and a 1.39 M� NS orbit in 7.8 hours. Also, the PSR B1534+12 system containstwo neutron stars, 1.33 and 1.135 M� respectively, in a 10 hour period. The PSR J0737-3039system also has two neutron stars, of 1.24 M� and 1.34 M�, in a 2.4 hour period (Kilic et.al, 2013).

4. Double White Dwarfs comprise the final category of this study. These compact systems arefar more abundant than any other source of gravitational waves radiation in the galaxy (Kilicet. al, 2013). We look at WD 0957-666, in which a 0.32 M� WD and a 0.37 M� WD orbitin 1.46 hours, and WD J0651, where a 0.26 M� and a 0.5 M� WDs orbit each other in 12.75minutes (Kilic et. al, 2013).

2 Overview of General Relativity and Gravitational Waves

Einstein developed General Relativity in an attempt to explain the force of gravity in lightof his discovery that the speed of light is constant in all frames of reference. In short, he foundthat spacetime is not flat; the presence of matter (energy) curves it and warps it (Miller, 2008).Objects that are spatially near large concentrations of matter or energy will move based on theresulting curvature of spacetime. Acceleration due to gravity, therefore, is simply the process ofan object moving along a path defined by these relativistic warps; its path of travel and its rate ofacceleration depend on the properties of the mass around it that caused the local spacetime to warp(Miller, 2008).

An important offshoot of gravitational warping is the idea that moving bodies produce fluctua-tions in the fabric of spacetime. An orbital system, for example, produces a rippling in spacetimethat propagates out and away from the system. The frequency of the system’s orbit determines

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the frequency of the emitted gravitational ripple, or wave (Miller, 2008). This is analogous to theway the frequency of an oscillating electric field determines the frequency of the emitted electro-magnetic wave. However, in the gravitational case, the masses of the bodies must very large andthe frequency of orbit must be very high for the wave to be detected with our current technology(Miller, 2008). For systems with ordinary masses and frequencies, the gravitational wave gener-ated will too weak to be detected. This is due a remarkable property of gravitational radiation: ithardly interacts with or disturbs ordinary matter as it passes by (Miller, 2008). Fortunately for ourpurpose, some extreme systems, such as ultra-compact binaries, move violently enough to distortspacetime to the extent that we may detect it, even if the wave source is far away from the Earth.

In consistency with the law of conservation of energy, any system that radiates energy must loseenergy at a rate equal to that at which it radiates. Verily, as has been supported empirically (seefigure 3, below), binary systems that emit energy in the form of gravitational waves simultaneouslylose gravitational potential energy (Miller, 2008). That is to say, as time progresses, the radius ofthe orbital system decreases until the two objects eventually collide, as shown below:

Figure 1: An artist’s depiction of two neutron stars in orbit (left). Their orbit generates gravita-tional waves (shown as the spiraling white crests), which, due to energy loss, cause a decrease in or-bital radius, which causes an inspiral (center), and eventually a coalescence explosion (right). (Image:NASA/CXC/GSFC/T.Strohmayer)

3 The LISA Mission

The foremost aspect of these ultra-compact binaries is that they emit gravitational waves.First predicted by Albert Einstein in 1916, gravitational waves are ripples in the curvature of space-time that propagate at the speed of light. They are emitted when two high mass objects producea gravitational tidal field that changes with time. This changing tidal field is known as gravita-tional radiation, and it manifests by squeezing and stretching the fabric of spacetime as it travels(Miller, 2008). Gravitational radiation can be detected using both Earth-based instruments – suchas the currently functional detector LIGO (Laser Interferometer Gravitational-Wave Observatory)– and space-based instruments, such as the proposed mission, LISA (Laser Interferometer SpaceAntenna). The LISA will consist of three spacecraft oriented in an equilateral triangle; each sideabout five million kilometers long, with a laser connecting each vertex. The entire configurationwill be launched into a heliocentric orbit (around the sun), similar to that of the Earth. The LISAwill be set up such that its lasers can detect passing gravitational waves using a technique knownas interferometry (Rowan, 2000). Under this technique, a gravitational wave traveling in a planeperpendicular to the laser triangle will slightly increase the length of one arm, while decreasing thelength of another, which allows us to calculate the total amplitude of the wave (Kokkotas, 2002).

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Figure 2: An illustration of the heliocentric orbit of the LISA detector (not to scale) (Image: Rowan, 2000)

Our key to understanding the evolution of ultra-compact binary systems lies in the detectionof gravitational wave emission. These waves transmit energy away from the binary systems, andcause a decay in their orbital periods. The LISA mission will allow us to detect wave informationthat was previously unavailable through electromagnetic observations.

4 Observation Techniques

The first direct observation of gravitational waves occurred in September 2015 and was an-nounced in February 2016. The wave was produced by two black holes in a binary system, of36 M� and 29 M� respectively, that coalesced to form a single black hole. The collision oc-curred about 1.3 billion lightyears away from Earth, but was violent enough to be detected by theEarth-based detector, LIGO.1

However, since the LISA mission is still just a future prospect, scientists are still unable todirectly detect GWs emanating from ultra-compact binary systems, as the frequency of these wavesis not as great as those emitted by colliding black holes (Mirshekari, 2016). The LIGO detectoris capable of detecting phenomena of this frequency, while the LISA is not (Mirshekari, 2016).So far, scientists have only been able to theorize (although with high certainty) that ultra-compactbinaries are emitting gravitational waves, due to the observed decay in the orbital period of thesystems:

Figure 3: This famous chart, displaying the shift in periastron time (corresponding to orbital period decay)vs. time for the Double Pulsar system PSR1913+16, or the Hulse-Taylor binary, strongly supports thehypothesis that binary systems emit gravitational waves. The data points for this system are shown alongwith the theoretical prediction of orbital decay (the thin line), from Einstein’s General Relativity; the twomatch perfectly. (Miller, 2008)

1Commisariat, LIGO Detects First Ever Gravitational Waves from Two Merging Black Holes

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The procedure to measure orbital period is called high-speed photometry. The instrument (suchas the 8.1 m Gemini North telescope) measures the magnitude of light emitted from a source,which, in this case, is usually the primary star; the more massive star in the system (Kilic et. al,2013). When the smaller orbital companion passes in front of the primary star, as seen from Earth,the magnitude intensity of light changes considerably. By measuring the interval of time betweeneach eclipse over an extended period of time, it is possible to detect a decay in the orbital period(Kilic et. al, 2013). See figure 2, below.

Figure 4: A graph of change in magnitude vs. orbital phase (time) for the double white dwarf system J0651.Using the dotted red line as reference, it is easy to see that the secondary star eclipses the primary soonerand sooner, indicating a decay in orbital period that is almost certainly due to the emission of gravitationalwaves. (Hermes et al. 2012b)

5 Description of the Calculations

For the calculations, I focused on the strain and frequency of the gravitational waves pro-duced by the ten ultra-compact binary systems described above. As these waves propagate out ofthe binary system, they carry away energy. This energy loss manifests as a gradual decrease inradius of the system. The rate at which the radius decreases depends on the masses of the twobodies (Kokkotas, 2002).

The purpose of the calculations below is to determine the strain (i.e. the dimensionless propertycorresponding to the amplitude) of the gravitational waves emitted by various types of compactbinary systems, and how this strain depends on the system’s frequency. Then I will determinewhether the LISA detector will be sensitive enough to detect the waves released by those systems.I will also determine how quickly the systems will lose energy and eventually collapse, and howthis property depends on the present frequency of the system. Furthermore, I will determine thegravitational luminosity of the systems, that is, the amount of energy released per second in theform of gravitational radiation, and how this value depends on system frequency.

5.1 Assumptions• To simplify the calculations, it is always assumed:

1. That the system is in perfectly circular orbit,

2. That the separation between the bodies in the system is large enough to treat the bodiesas point sources,

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3. That the detector is detecting from a location exactly in the plane of the system’s orbit(Kokkotas, 2002).

These assumptions may not be realistic for most of the ultra-compact binary systems in the MilkyWay, or for detectors on Earth – however, without these assumptions, the calculations becomeexceedingly difficult.

6 Data

Figure 5: The different colors represent the four different types of compact binary systems, in the orderthat they were described in section one: (Orange = AM CVn, Yellow = X-Ray Binary, Teal = Double Pulsar,Pink = Double WD)

6.1 Description of the New Columns:• Frequency is simply the inverse of the period: f = 1/P . The unit is hertz, or 1/seconds.

• m1 and m2 were determined by multiplying the given fractional sun-masses of the systemsby the mass of the sun, which is 1.989x1030 kg.

• M is the combined mass of the system in kg, or M = m1 +m2 (Kokkotas, 2002).

• µ is the reduced mass of the system in kg, given by m1m2/M (Kokkotas, 2002).

• r is the radius to the system, or the distance from the system to Earth that the wave musttravel from source to observer/detector. This is a given value, in Megaparsecs.

• a is the semimajor axis of the orbital system in meters. In our case, however, since we treateach system as maintaining a perfectly circular orbit, we can treat a as the radius of theorbit. This is calculated using the period P and Kepler’s third law, which states that a =(P 2GM/(4pi2))1/3

• da/dt is the time rate of change in orbital radius, in meters per second. The formula isda/dt = −(64G3µM2)/(5c5a3), where c is the speed of light in meters per second (Kokko-tas, 2002). Note that all values should and will be negative because the semi-major axes (ororbital radii, for circular orbits) of these systems decrease with time.

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• τ is the coalescence time (in years). This is the time left until the system loses enoughenergy to gravitational radiation that the radius shrinks small enough so that the two stellarcomponents actually collide with each other. The formula for τ (in seconds) is obtainedfrom manipulating the above formula for da/dt, integrating, and solving for τ , to give τ =5c5a4/(256µM2G3). That value is then easily converted into years, to obtain a more relevantand interesting number.(Kokkotas, 2002).

• h, or strain, is the dimensionless property of the emitted gravitational wave that correspondsto its amplitude. In essence, it is the fractional change or fluctuation of a unit amount (dis-tance) of spacetime as the gravitational wave passes by.

h = 5× 10−22(M/2.8M�)2/3(µ/0.7M�)(f/100Hz)

2/3(15Mpc/r) (Kokkotas, 2002)

• LGW , or gravitational luminosity, is a measure of the amount of energy per second emitted bythe system along with the gravitational radiation. Although this is not energy that we can har-ness or observe in the conventional sense, it exists nonetheless. LGW = (32G4M3µ2)/(5c5a5)(Kokkotas, 2002).

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7 Charts, Analysis, and Conclusions

Figure 6: In this chart, as in the data table, the color of the dot corresponds to the type of binary. (Orange= AM CVn, Yellow = X-Ray Binary, Teal = Double Pulsar, Pink = Double WD). Note that systems abovethe LISA sensitivity line are detectable by LISA, while systems below the line are not massive enough/donot have a high enough frequency to be detected. (Credit for LISA sensitivity curves: Mirshekari, 2016.Converted from an image by the app WebPlot Digitizer at http://arohatgi.info/WebPlotDigitizer/).

In Figure 6, we can see that data points are relatively clustered by binary type, suggest-ing that each type of binary system emits gravitational waves with a characteristic range of fre-quency and strain, to a limited extent. Therefore, if we wish to obtain the full picture of compactbinary-emitted gravitational waves in the future, it will be necessary to observe as many systemsas possible for all of the categories of compact binary.

Two of the points in Figure 6, corresponding to the X-Ray binary system RX J0806.3+1527and the AM CVn system HM Cnc, lie above the LISA sensitivity line – that is, once launched,the LISA detector will be sensitive enough to detect the gravitational waves emitted from thissystem. However, both HM Cnc and the prototype AM CVn (the two points shown in orange) areLISA verification binaries, that is, they are supposed to be above the line, so that when the LISAis launched, it can observe those systems and either verify or reject the theoretical predictionsmade about gravitational radiation from compact binary systems (Kilic et al, 2013). However, onthis chart, prototype AM CVn (the orange dot below the LISA sensitivity limit) appears belowthe LISA sensitivity line. This error could be due to the assumptions made about circular orbits,separation of bodies, or planar observation. Because the orbital frequencies of all of the systems

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were given values, any error must lie in the calculation of h, characteristic strain. The prototypeAM CVn ought to lie above the LISA limit, and we were given its frequency, thus, by looking atits location on the chart in relation to the LISA limit at the same frequency, its calculated value forstrain must be too low.

Recalling that AM CVn stars are defined by a transferring of mass between the two componentstars, it becomes clear that the assumption to treat the two bodies as point sources is mistaken. Inaddition, any inclination in the plane of orbit of the system, as observed from Earth, would seemto decrease the amplitude of the wave, because the strongest part of the wave would no longerbe traveling in a plane perpendicular to the interferometer. Instead, we would detect a fraction ofthe wave’s full amplitude, the value of which depends on the degree of the inclination (Kokkotas,2002). Lastly, if we take the systems’ elliptical orbits consideration, more complications arise.In this case, due to the increasing and decreasing orbital velocities of the stars as they progressthrough their phases, the amplitude of the emitted gravitational wave might fluctuate in time.

Assumption and error aside, this chart illustrates an important caveat in the study of gravita-tional waves: Our current understanding of technology simply does not allow us to detect all ofthe gravitational waves in the universe, even those emitted from some of the most extreme systemsin our Galaxy. Our first steps into gravitational wave astronomy will provide extraordinary newinformation, but it is important to realize that we are only accessing a small fraction of the entirepicture. Even after the LISA and other proposed detectors are launched and functional, we must bethorough and fastidious with the data, to ensure that we do not jump to unjustified or misleadingconclusions.

Figure 7:

Figure 7 suggests that there is an exponential relationship between a system’s orbital radiusrate of change and its time left until coalescence. Specifically, looking at the chart, the trendsuggests that as the rate of change in orbital radius increases, the time until coalescence decreasesat an increasing rate. This makes intuitive sense, based on the nature of changing rates. In addition,using this chart and a basic knowledge of binary kinematics, it becomes clear that as the two stars

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get closer and closer together, they begin to orbit each other faster and faster, producing higherfrequency gravitational waves, which carry off a greater amount energy per second, which in turncauses the orbital radius to decrease quicker and quicker. It is a positive feedback loop that speedsup the system as it progresses, and results in a catastrophic explosion.

Figure 8:

Figure 8 suggests that there may be an exponential relationship between frequency of a systemand the gravitational luminosity output of that system. As frequency increases, luminosity in-creases at an increasing rate. This relation clarifies the reason why gravitational wave astronomersseek out high-frequency systems – they release exponentially more energy than low-frequencysystems, making them much more interesting and easier to detect than low-frequency systems.Another noteworthy aspect of this chart is the magnitude of the luminosities of each system. Sir-ius, the brightest star in the night sky, has a luminosity of 1027 watts. The luminosity of themost luminous system is only about one order of magnitude lower! That is to say, if our eyes hadevolved to detect the energy from gravitational radiation as well as from electromagnetic radiation,the night sky would appear significantly brighter (Miller, 2008).

8 Future Implications

Since antiquity, until this very year, astronomy has been a “one-sense” field. That is to say,nearly all of our knowledge of the universe has come from the detection of the electromagneticwaves emitted by astronomical objects and events. For this reason, our developing ability to detectgravitational waves is revolutionary. With this new ability, we will receive information from objectsand events that have heretofore been invisible to us. The LISA and the LIGO detectors, as well asothers, will help us to map the distribution of ultra-compact binaries in our Galaxy, in addition toother, more rare objects within and beyond our Galaxy.

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The first detection of gravitational waves is analogous to a deaf man who obtains the ability tohear for the first time. For his whole life, he has only gained information about the world using hissense of sight. Now that he can hear, his sense of the external world is heightened exponentially.This analogy relates exactly to astronomers “listening” to gravitational waves. The new detectorswill open our ears to the universe. We will gain new knowledge and insight into the darkness thatsurrounds us.

Specifically, in the coming years, as we continue to gather data from ultra-compact binary starsand star systems, supernovae, and black holes, we will obtain a better understanding of the mostextreme events our universe has to offer. This information will serve to support or reject currenttheories, and, perhaps more importantly, it will provide data with which we might construct newtheories and establish a more complete understanding of the laws of the universe. Gravity hasalways been a baffling concept for our species, as we are yet unable to reconcile its existence withthe equally complicated theory of quantum mechanics. For example, we do not know what occursin the center of a black hole; nor have we been able to model the structure of a neutron star’sdense inner core. Also, the existence of the graviton – a massless particle, hypothesized to beresponsible for the effects of gravity – still awaits empirical support. The study of gravitationalwaves may give us a more detailed description of these phenomena. Finally, study of the cosmicmicrowave background has provided evidence that suggests our universe began 13.8 billion yearsago with the Big Bang. Detection of the gravitational waves produced during the explosion – orsoon thereafter, during the inflationary period – could lead us closer to an understanding of howand why the universe came to be in the first place.2

There is an abundance of gravitational wave sources in the observable universe, the most abun-dant of which are ultra-compact binary systems (Kokkotas, 2002). Once we develop and launchthe technology to properly detect waves from a wide variety of sources, we will have access to thewealth of information that sits all around us. Furthermore, the most special property of gravita-tional waves is that their propagation through spacetime is relatively unhindered by the presenceof interstellar gas and dust in their path. Whereas light waves are often weakened or even com-pletely absorbed by such material, gravitational waves, for the most part, tend to push right through(Kokkotas, 2002). The implication here is twofold: Not only will we be able to detect gravitationalwaves from a wide variety of sources and areas throughout the universe, but we can also be rea-sonably sure (upon inspection of the path) that those waves did not radically change form due tosome random obstruction during their trip to Earth.

Although we cannot know for certain what gravitational wave detection will show us as wemove forward, it is bound to change the way we think about the universe. Of course, there is achance that we end up with a plethora of useless data about ultra-compact binaries, black holes andsupernovae. However, there is also a chance that we make a revolutionary breakthrough and changethe course of science forever. It almost evokes a sense of adventurousness, as we branch out intoa new realm of study, with no certainty of the results and phenomena we will uncover. Whether itproduces enlightening results or not, the study and analysis of gravitational wave emitters is certainto make an impact on astronomical physics for the foreseeable future.

2Kramer, Gravitational Waves: The Big Bang’s Smoking Gun.

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References

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[2] Hermes, J. J. et. al. 2012b, ArXiv e-prints. 1208.5051

[3] Kilic, M., et. al. 2013, ASPC 467, Ultra-Compact Binaries: eLISA Verification Sources, ed.G. Auger, P. Binetruy and E. Plagnol (Paris, France:ASP)

[4] Kokkotas, K. 2002, Encyclopedia of Physical Science and Technology, 3rd ed, vol. 7

[5] Kramer, Miriam. “Gravitational Waves: The Big Bang’s Smoking Gun.” Space.com. N.p., 11Feb. 2016. Web. 24 May 2016.

[6] Miller, Cole. 2008. Overview of General Relativity and Gravitational Waves. Depart-ment of Astronomy, University of Maryland. ASTR 498, High Energy Astrophysics.<https://www.astro.umd.edu/ miller/teaching/astr498/>

[7] Mirshekari, Saeed. “Plotting the Sensitivity Curves of Gravitational Waves De-tectors.” Dr Manhattans Diary. Wordpress, 28 Apr. 2014. Web. 24 May 2016.<https://smirshekari.wordpress.com/2014/04/28/plotting-the-sensitivity-curves-of-gravitational-waves-detectors/>

[8] Nelemans, G., et. al. 2010. The Astrophysics of Ultra-Compact Binaries. ASTRO2010 DecadalReview.

[9] Rowan, Sheila. “Gravitational Wave Detection by Interferometry (Ground and Space).”Living Views in Relativity (2000). Web. 24 May 2016. <http://hermes.roua.org/hermes-pub/lrr/2000/3/article.xhtml>

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