There are several types of ultra-compact binary star systems that orbit each other with periods of less than an hour. These systems emit gravitational waves due to their strong gravitational fields changing over time. The Laser Interferometer Space Antenna (LISA) mission aims to detect these gravitational waves. While current ground-based detectors cannot detect the waves from ultra-compact binaries, LISA may be able to do so due to observing from space. The document provides data on four example binary systems and calculates their orbital decay rates and the strain of the gravitational waves emitted.

1. Ultra-Compact Binary Systems and the Resulting Gravitational
Waves
Sam Carey
Term 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 that
consist of white dwarfs, neutron stars, and other small, dense stars. An ultra-compact binary is
defined as a binary system with a very short orbital period: generally, less than one hour. When
the system is sufficiently massive, and has a high enough frequency, it will emit gravitational
waves large enough to detect. However, only a few of these systems produce waves strong
enough 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 emitting
systems, display calculations of certain properties of the systems and the gravitational waves
they emit, and explain the future potential to advance our knowledge of the universe based on
the study of these systems.
1 System Classification
There are a number of types of ultra-compact binary systems, differing based on the physical
make up of their orbiting stars. Due to their high frequency, and thus larger amplitude gravitational
waves (see section Data, below, for an explanation of gravitational wave amplitude), scientists
choose to examine the systems in which the two stars orbit each other in the smallest amount
of time. These high-energy systems are generally assumed to produce waves large enough to be
detected by human technology. In the table in the Data section, there are data from four different
types 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 as
a 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 dramatically
in brightness before returning to a dimmer state (Kilic et. al., 2013). Specifically, when the
primary star accretes a critical amount of hydrogen from the secondary star, the density and
temperature 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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2. 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 Canum
Venaticorum, which consists of a 0.71 M WD and a 0.13 M companion orbiting in a 17
minute 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 another
normal star called the donor. The high mass accretor steals mass away from the lower mass
donor as the donor orbits the accretor. The accretor can be a neutron star, ie, the smallest and
most dense type of star known in the universe. Neutron stars can have radii as small as 11
kilometers, and masses of up to twice that of the sun (Kilic et. al, 2013). They result from
the gravitational collapse of a heavy star, following that star’s termination of fusion and the
resulting supernova. This type of system is named for the luminous X-rays that are emitted
as a result of the accretion process. The systems in this study are 4U 1820-30, where a 1.4
M NS and a 0.06 M WD orbit in a 685 second period; 4U 01513-40, where a 1.4 M NS
and a 0.05 M WD orbit in 17 minutes; and RX J0806.3+1527, where two 0.5 M WDs
orbit in 5.3 minutes (Kilic et. al, 2013).
3. Double Pulsar Systems have two extremely dense, pulsing neutron stars in orbit around
each other. This study looks at the Hulse-Taylor Binary Pulsar, PSR B1913+16, in which
1.44 M NS and a 1.39 M NS orbit in 7.8 hours. Also, the PSR B1534+12 system contains
two neutron stars, 1.33 and 1.135 M respectively, in a 10 hour period. The PSR J0737-3039
system 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 are
far more abundant than any other source of gravitational waves radiation in the galaxy (Kilic
et. al, 2013). We look at WD 0957-666, in which a 0.32 M WD and a 0.37 M WD orbit
in 1.46 hours, and WD J0651, where a 0.26 M and a 0.5 M WDs orbit each other in 12.75
minutes (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 light
of his discovery that the speed of light is constant in all frames of reference. In short, he found
that 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 the
resulting curvature of spacetime. Acceleration due to gravity, therefore, is simply the process of
an object moving along a path defined by these relativistic warps; its path of travel and its rate of
acceleration 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 spacetime
that propagates out and away from the system. The frequency of the system’s orbit determines
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3. the frequency of the emitted gravitational ripple, or wave (Miller, 2008). This is analogous to the
way 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 and
the 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: it
hardly interacts with or disturbs ordinary matter as it passes by (Miller, 2008). Fortunately for our
purpose, some extreme systems, such as ultra-compact binaries, move violently enough to distort
spacetime 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 lose
energy at a rate equal to that at which it radiates. Verily, as has been supported empirically (see
figure 3, below), binary systems that emit energy in the form of gravitational waves simultaneously
lose gravitational potential energy (Miller, 2008). That is to say, as time progresses, the radius of
the 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 produce
a 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 – such
as the currently functional detector LIGO (Laser Interferometer Gravitational-Wave Observatory)
– and space-based instruments, such as the proposed mission, LISA (Laser Interferometer Space
Antenna). The LISA will consist of three spacecraft oriented in an equilateral triangle; each side
about five million kilometers long, with a laser connecting each vertex. The entire configuration
will be launched into a heliocentric orbit (around the sun), similar to that of the Earth. The LISA
will be set up such that its lasers can detect passing gravitational waves using a technique known
as interferometry (Rowan, 2000). Under this technique, a gravitational wave traveling in a plane
perpendicular to the laser triangle will slightly increase the length of one arm, while decreasing the
length of another, which allows us to calculate the total amplitude of the wave (Kokkotas, 2002).
3

4. 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 detection
of gravitational wave emission. These waves transmit energy away from the binary systems, and
cause a decay in their orbital periods. The LISA mission will allow us to detect wave information
that 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, of
36 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 the
Earth-based detector, LIGO.1
However, since the LISA mission is still just a future prospect, scientists are still unable to
directly detect GWs emanating from ultra-compact binary systems, as the frequency of these waves
is not as great as those emitted by colliding black holes (Mirshekari, 2016). The LIGO detector
is 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-compact
binaries are emitting gravitational waves, due to the observed decay in the orbital period of the
systems:
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 the
hypothesis that binary systems emit gravitational waves. The data points for this system are shown along
with the theoretical prediction of orbital decay (the thin line), from Einstein’s General Relativity; the two
match perfectly. (Miller, 2008)
1
Commisariat, LIGO Detects First Ever Gravitational Waves from Two Merging Black Holes
4

5. The procedure to measure orbital period is called high-speed photometry. The instrument (such
as 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 between
each 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 sooner
and sooner, indicating a decay in orbital period that is almost certainly due to the emission of gravitational
waves. (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 of
the binary system, they carry away energy. This energy loss manifests as a gradual decrease in
radius of the system. The rate at which the radius decreases depends on the masses of the two
bodies (Kokkotas, 2002).
The purpose of the calculations below is to determine the strain (i.e. the dimensionless property
corresponding to the amplitude) of the gravitational waves emitted by various types of compact
binary systems, and how this strain depends on the system’s frequency. Then I will determine
whether 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 how
this property depends on the present frequency of the system. Furthermore, I will determine the
gravitational luminosity of the systems, that is, the amount of energy released per second in the
form 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 bodies
as point sources,
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6. 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 Milky
Way, or for detectors on Earth – however, without these assumptions, the calculations become
exceedingly difficult.
6 Data
Figure 5: The different colors represent the four different types of compact binary systems, in the order
that 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 systems
by 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 must
travel 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 treat
each system as maintaining a perfectly circular orbit, we can treat a as the radius of the
orbit. This is calculated using the period P and Kepler’s third law, which states that a =
(P2
GM/(4pi2
))1/3
• da/dt is the time rate of change in orbital radius, in meters per second. The formula is
da/dt = −(64G3
µM2
)/(5c5
a3
), 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 (or
orbital radii, for circular orbits) of these systems decrease with time.
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7. • τ is the coalescence time (in years). This is the time left until the system loses enough
energy to gravitational radiation that the radius shrinks small enough so that the two stellar
components actually collide with each other. The formula for τ (in seconds) is obtained
from manipulating the above formula for da/dt, integrating, and solving for τ, to give τ =
5c5
a4
/(256µM2
G3
). That value is then easily converted into years, to obtain a more relevant
and interesting number.(Kokkotas, 2002).
• h, or strain, is the dimensionless property of the emitted gravitational wave that corresponds
to 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 by
the 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
= (32G4
M3
µ2
)/(5c5
a5
)
(Kokkotas, 2002).
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8. 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 above
the LISA sensitivity line are detectable by LISA, while systems below the line are not massive enough/do
not 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 compact
binary-emitted gravitational waves in the future, it will be necessary to observe as many systems
as 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+1527
and 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 this
system. However, both HM Cnc and the prototype AM CVn (the two points shown in orange) are
LISA verification binaries, that is, they are supposed to be above the line, so that when the LISA
is launched, it can observe those systems and either verify or reject the theoretical predictions
made about gravitational radiation from compact binary systems (Kilic et al, 2013). However, on
this chart, prototype AM CVn (the orange dot below the LISA sensitivity limit) appears below
the 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
8

9. were given values, any error must lie in the calculation of h, characteristic strain. The prototype
AM CVn ought to lie above the LISA limit, and we were given its frequency, thus, by looking at
its location on the chart in relation to the LISA limit at the same frequency, its calculated value for
strain must be too low.
Recalling that AM CVn stars are defined by a transferring of mass between the two component
stars, it becomes clear that the assumption to treat the two bodies as point sources is mistaken. In
addition, any inclination in the plane of orbit of the system, as observed from Earth, would seem
to decrease the amplitude of the wave, because the strongest part of the wave would no longer
be traveling in a plane perpendicular to the interferometer. Instead, we would detect a fraction of
the 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 progress
through 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 of
the gravitational waves in the universe, even those emitted from some of the most extreme systems
in our Galaxy. Our first steps into gravitational wave astronomy will provide extraordinary new
information, but it is important to realize that we are only accessing a small fraction of the entire
picture. Even after the LISA and other proposed detectors are launched and functional, we must be
thorough and fastidious with the data, to ensure that we do not jump to unjustified or misleading
conclusions.
Figure 7:
Figure 7 suggests that there is an exponential relationship between a system’s orbital radius
rate of change and its time left until coalescence. Specifically, looking at the chart, the trend
suggests that as the rate of change in orbital radius increases, the time until coalescence decreases
at 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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10. get closer and closer together, they begin to orbit each other faster and faster, producing higher
frequency gravitational waves, which carry off a greater amount energy per second, which in turn
causes the orbital radius to decrease quicker and quicker. It is a positive feedback loop that speeds
up 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 system
and 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 astronomers
seek out high-frequency systems – they release exponentially more energy than low-frequency
systems, 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 the
most luminous system is only about one order of magnitude lower! That is to say, if our eyes had
evolved 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 electromagnetic
waves emitted by astronomical objects and events. For this reason, our developing ability to detect
gravitational waves is revolutionary. With this new ability, we will receive information from objects
and events that have heretofore been invisible to us. The LISA and the LIGO detectors, as well as
others, will help us to map the distribution of ultra-compact binaries in our Galaxy, in addition to
other, more rare objects within and beyond our Galaxy.
10

11. The first detection of gravitational waves is analogous to a deaf man who obtains the ability to
hear for the first time. For his whole life, he has only gained information about the world using his
sense 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 detectors
will open our ears to the universe. We will gain new knowledge and insight into the darkness that
surrounds us.
Specifically, in the coming years, as we continue to gather data from ultra-compact binary stars
and star systems, supernovae, and black holes, we will obtain a better understanding of the most
extreme events our universe has to offer. This information will serve to support or reject current
theories, and, perhaps more importantly, it will provide data with which we might construct new
theories and establish a more complete understanding of the laws of the universe. Gravity has
always been a baffling concept for our species, as we are yet unable to reconcile its existence with
the equally complicated theory of quantum mechanics. For example, we do not know what occurs
in the center of a black hole; nor have we been able to model the structure of a neutron star’s
dense inner core. Also, the existence of the graviton – a massless particle, hypothesized to be
responsible for the effects of gravity – still awaits empirical support. The study of gravitational
waves may give us a more detailed description of these phenomena. Finally, study of the cosmic
microwave background has provided evidence that suggests our universe began 13.8 billion years
ago with the Big Bang. Detection of the gravitational waves produced during the explosion – or
soon thereafter, during the inflationary period – could lead us closer to an understanding of how
and 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 launch
the technology to properly detect waves from a wide variety of sources, we will have access to the
wealth 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 presence
of 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 gravitational
waves 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 to
some random obstruction during their trip to Earth.
Although we cannot know for certain what gravitational wave detection will show us as we
move forward, it is bound to change the way we think about the universe. Of course, there is a
chance that we end up with a plethora of useless data about ultra-compact binaries, black holes and
supernovae. However, there is also a chance that we make a revolutionary breakthrough and change
the course of science forever. It almost evokes a sense of adventurousness, as we branch out into
a new realm of study, with no certainty of the results and phenomena we will uncover. Whether it
produces enlightening results or not, the study and analysis of gravitational wave emitters is certain
to make an impact on astronomical physics for the foreseeable future.
2
Kramer, Gravitational Waves: The Big Bang’s Smoking Gun.
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