In
astrophysics the chirp mass of a compact binary system determines the
leading-order The leading-order terms (or corrections) within a mathematical equation, expression or model are the terms with the largest order of magnitude.J.K.Hunter, ''Asymptotic Analysis and Singular Perturbation Theory'', 2004. http://www.math.ucdavis.edu/ ...
orbital evolution of the system as a result of energy loss from emitting
gravitational waves
Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
. Because the gravitational wave frequency is determined by orbital frequency, the chirp mass also determines the frequency evolution of the gravitational wave signal emitted during a binary's
inspiral
Orbital decay is a gradual decrease of the distance between two orbiting bodies at their closest approach (the periapsis) over many orbital periods. These orbiting bodies can be a planet and its satellite, a star and any object orbiting it, or c ...
phase. In
gravitational wave data analysis it is easier to measure the chirp mass than the two component masses alone.
Definition from component masses
A two-body system with component masses
and
has a chirp mass of
:
The chirp mass may also be expressed in terms of the total mass of the system
and other common mass parameters:
* the
reduced mass
In physics, the reduced mass is the "effective" Mass#Inertial mass, inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, how ...
:
*:
* the mass ratio
:
*:
or
* the symmetric mass ratio
:
*:
*: The symmetric mass ratio reaches its maximum value
when
, and thus
* the geometric mean of the component masses
:
*:
*: If the two component masses are roughly similar, then the latter factor is close to
so
. This multiplier decreases for unequal component masses but quite slowly. E.g. for a 3:1 mass ratio it becomes
, while for a 10:1 mass ratio it is
Orbital evolution
In
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, the phase evolution of a
binary orbit can be computed using a
post-Newtonian expansion
In general relativity, the post-Newtonian expansions (PN expansions) are used for finding an approximate solution of the Einstein field equations for the metric tensor. The approximations are expanded in small parameters which express orders of ...
, a perturbative expansion in powers of the orbital velocity
. The first order gravitational wave frequency,
, evolution is described by the
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
:
,
where
and
are the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
and
Newton's gravitational constant, respectively.
If one is able to measure both the frequency
and frequency derivative
of a gravitational wave signal, the chirp mass can be determined.
To disentangle the individual component masses in the system one must additionally measure higher order terms in the post-Newtonian expansion.
Mass-redshift degeneracy
One limitation of the chirp mass is that it is affected by
redshift
In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
; what is actually derived from the observed gravitational waveform is the product
:
where
is the redshift. This redshifted chirp mass is larger than the source chirp mass, and can only be converted to a source chirp mass by finding the redshift
.
This is usually resolved by using the observed amplitude to find the chirp mass divided by distance, and solving both equations using
Hubble's law
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving ...
to compute the relationship between distance and redshift.
Xian Chen has pointed out that this assumes non-cosmological redshifts (
peculiar velocity
Peculiar motion or peculiar velocity refers to the velocity of an object relative to a ''rest frame'' — usually a frame in which the average velocity of some objects is zero.
Galactic astronomy
In galactic astronomy, peculiar motion refers to ...
and
gravitational redshift
In physics and general relativity, gravitational redshift (known as Einstein shift in older literature) is the phenomenon that electromagnetic waves or photons travelling out of a gravitational well (seem to) lose energy. This loss of energ ...
) are negligible, and questions this assumption.
If a binary pair of
stellar-mass black hole
A stellar black hole (or stellar-mass black hole) is a black hole formed by the gravitational collapse of a star. They have masses ranging from about 5 to several tens of solar masses. The process is observed as a hypernova explosion or as a g ...
s merge while closely orbiting a
supermassive black hole
A supermassive black hole (SMBH or sometimes SBH) is the largest type of black hole, with its mass being on the order of hundreds of thousands, or millions to billions of times the mass of the Sun (). Black holes are a class of astronomical obj ...
(an
extreme mass ratio inspiral
In astrophysics, an extreme mass ratio inspiral (EMRI) is the orbit of a relatively light object around a much heavier (by a factor 10,000 or more) object, that gradually spirals in due to the emission of gravitational waves. Such systems are lik ...
), the observed gravitational wave would experience significant gravitational and doppler redshift, leading to a falsely low redshift estimate, and therefore a falsely high mass. He suggests that there are plausible reasons to suspect that the SMBH's
accretion disc
An accretion disk is a structure (often a circumstellar disk) formed by diffuse material in orbital motion around a massive central body. The central body is typically a star. Friction, uneven irradiance, magnetohydrodynamic effects, and other f ...
and
tidal force
The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenomen ...
s would enhance the merger rate of black hole binaries near it, and the consequent falsely high mass estimates would explain the unexpectedly large masses of
observed black hole mergers. (The question would be best resolved by a lower-frequency gravitational wave detector such as
LISA Lisa or LISA may refer to:
People
People with the mononym
* Lisa Lisa (born 1967), American actress and lead singer of the Cult Jam
* Lisa (Japanese musician, born 1974), stylized "LISA", Japanese singer and producer
* Lisa Komine (born 1978), J ...
which could observe the EMRI waveform.)
See also
*
Reduced mass
In physics, the reduced mass is the "effective" Mass#Inertial mass, inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, how ...
*
Two-body problem in general relativity
The two-body problem in general relativity is the determination of the motion and gravitational field of two bodies as described by the field equations of general relativity. Solving the Kepler problem is essential to calculate the bending of lig ...
Note
References
{{Gravitational waves
-
Gravitational-wave astronomy