1. Ultra-Compact Binaries and the Resulting Gravitational Waves
Sam Carey
Astronomy 15 Term Project
Spring 2016
An ultra-compact binary is a star system, consisting of dense
White Dwarfs or Neutron Stars, with a very short orbital period –
generally less than one hour. When the system is sufficiently
massive, and has a high enough frequency, it emits gravitational
waves (GWs) large enough for us to detect. The proposed LISA
mission will enable the detection of such GWs. We present an
overview of GWs, the techniques to detect them, the properties
of some of the strongest emitting systems, and charts of certain
properties of both the systems and the GWs they emit.
Abstract
2. General Relativity and Gravitational Waves
Einstein’s General Relativity posits that the fabric of spacetime is not
flat; rather, in the presence of matter, it curves and warps. Massive,
moving bodies, such as co-orbiting binary stars, can therefore
produce ripples in spacetime – gravitational waves – that stretch
and squeeze spacetime as they propagate away from a source at the
speed of light. GWs carry energy out of the system, resulting in a
progressive decrease in orbital radius of the binary, and eventually a
collision of the two stars (Figure 1). Indeed, Einstein’s theory has
been corroborated through observations of decreases in binary
orbital radius, most famously in the Hulse-Taylor system (Figure 2).
Figure 1: An artist’s depiction of an inspiraling binary system and the
gravitational waves it emits (shown as spiraling white crests).
Figure 2: Period shift vs.
epoch for The Hulse-Taylor
binary system PSR1913+16.
The circular dots represent
empirical observations, the
dotted line represents
Einstein’s predictions from
General Relativity. It appears
that Einstein had it right.
3. The LISA Mission
The physical amount by which GWs squeeze spacetime,
even from GWs emitted by high mass/high frequency
binary stars, is miniscule. Thus, we need extremely
large instruments to detect them. The proposed LISA
mission is a space-based interferometer, set up in an
equilateral triangle with sides of ~5x106 km. The lasers
connecting each side will increase or decrease in length
as a GW passes by, enabling us to detect small
spacetime fluctuations.
Figure 3: An
illustration of LISA’s
(the red triangle)
configuration and
heliocentric orbit.
4. Data
Figure 4: Data from 10 GW emitting ultra-compact binary systems in
the Milky Way. The component masses are given in terms of our
sun’s mass. r is the distance from the system to Earth. The colors
correspond to the type of binary system: Orange = the cataclysmic
variable AM CVn stars, Yellow = X-Ray Binaries, Teal = Double
Pulsars, and Pink = Double White Dwarfs.
5. Charts/Conclusions
Using this data and the equations given in (Kokkotas, 2002), for
each system, we can calculate the strain, h (i.e., the fractional
change/squeeze in a unit distance of spacetime as the GW passes
by), the orbital radius rate of change, da/dt, the coalescence
time, τ (amount of time left until the stars collide), and the
gravitational luminosity, LGW (the amount of energy emitted
from the system, per second, in the form of gravitational
radiation).
6. References
Clockwise from top left: Figures
5, 6, and 7. In these figures, each
data point corresponds to one of
the ten binary systems listed
above. Points in figure 5 follow
the same color scheme as the
data table. Here, systems above
the green line can be detected by
LISA, those below are too faint.
Figure 6 displays the exponential
relationship between orbital
radius rate of change and
coalescence time. Figure 7
displays the relationship
between system frequency and
gravitational luminosity output,
thus clarifying why scientists seek
to observe the quickest-orbiting
binary systems: luminosity
increases exponentially with
frequency.
Hermes et al. 2012b, ArXiv, 1208.5051
Kilic et al. 2013, ASPC 467
Kokkotas. 2002, EPST, 3, 7
Mirshekari. 2014, Dr. Manhattan’s Diary
Nelemans et al. 2010, ASTRO2010 Decadal Review
Rowan, Living Views In Relativity, 2000