of 17
Original Paper
Ann. Phys. (Berlin)
529
, No. 1–2, 1600209 (2017) / DOI 10.1002/andp.201600209
The basic physics of the binary black hole merger GW150914
LIGO Scientific and VIRGO Collaborations
,
∗∗
Received 5 August 2016, revised 21 September 2016, accepted 22 September 2016
Published online 4 October 2016
The first direct gravitational-wave detection was made by
the Advanced Laser Interferometer Gravitational Wave Ob-
servatory on September 14, 2015. The GW150914 signal was
strong enough to be apparent, without using any wave-
form model, in the filtered detector strain data. Here, fea-
tures of the signal visible in the data are analyzed using
concepts from Newtonian physics and general relativity, ac-
cessible to anyone with a general physics background. The
simple analysis presented here is consistent with the fully
general-relativistic analyses published elsewhere, in show-
ing that the signal was produced by the inspiral and subse-
quent merger of two black holes. The black holes were each
of approximately
35 M

, still orbited each other as close as
350
km apart and subsequently merged to form a single
black hole. Similar reasoning, directly from the data, is used
to roughly estimate how far these black holes were from
the Earth, and the energy that they radiated in gravitational
waves.
1 Introduction
Advanced LIGO made the first observation of a gravi-
tational wave (GW) signal, GW150914 [1], on Septem-
ber 14th, 2015, a successful confirmation of a prediction
by Einstein’s theory of general relativity (GR). The sig-
nal was clearly seen by the two LIGO detectors located
in Hanford, WA and Livingston, LA. Extracting the full
information about the source of the signal requires de-
tailed analytical and computational methods (see [2–6]
and references therein for details). However, much can
be learned about the source by direct inspection of the
detector data and some basic physics [7], accessible to a
general physics audience, as well as students and teach-
ers. This simple analysis indicates that the source is two
black holes (BHs) orbiting around one another and then
merging to form another black hole.
A black hole is a region of space-time where the
gravitational field is so intense that neither matter nor
radiation can escape. There is a natural “gravitational ra-
dius” associated with a mass
m
, called the Schwarzschild
radius, given by
r
Schwarz
(
m
)
=
2
Gm
c
2
=
2
.
95

m
M


km
,
(1)
where M

=
1
.
99
×
10
30
kg is the mass of the Sun,
G
=
6
.
67
×
10
11
m
3
/
s
2
kg is Newton’s gravitational constant,
and
c
=
2
.
998
×
10
8
m
/
s is the speed of light. According
to the hoop conjecture, if a non-spinning mass is com-
pressed to within that radius, then it must form a black
hole [8]. Once the black hole is formed, any object that
comes within this radius can no longer escape out of it.
Here, the result that GW150914 was emitted by the in-
spiral and merger of two black holes follows from (1) the
strain data visible at the instrument output, (2) dimen-
sional and scaling arguments, (3) primarily Newtonian
orbital dynamics and (4) the Einstein quadrupole for-
mula for the luminosity of a gravitational wave source.
1
These calculations are straightforward enough that they
can be readily verified with pencil and paper in a short
time. Our presentation is by design approximate, empha-
sizing simple arguments.
Specifically, while the orbital motion of two bodies
is approximated by Newtonian dynamics and Kepler’s
laws to high precision at sufficiently large separations
and sufficiently low velocities, we will invoke Newtonian
dynamics to describe the motion even toward the end
point of orbital motion (We revisit this assumption in
lvc.publications@ligo.org
∗∗
Full author list appears at the end.
This is an open access article under the terms of the Creative
Commons Attribution License,which permits use,distribution
and reproduction in any medium,provided the original work is
properly cited.
1
In the terminology of GR corrections to Newtonian dynamics,(3) &
(4) constitute the “0th post-Newtonian”approximation (0PN) (see
Sec.4.4).A similar approximation was used for the first analysis of
binary pulsar PSR 1913
+
16 [9,10].
C

2016 The Authors.
Annalen der Physik
published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim
(1 of 17) 1600209
Original Paper
B. P. Abbot et al.: The basic physics of the binary black hole merger GW150914
Sec. 4.4). The theory of general relativity is a fully non-
linear theory, which could make any Newtonian analysis
wholly unreliable; however, solutions of Einstein’s
equations using numerical relativity (NR) [11–13] have
shown that a binary system’s departures from Newtonian
dynamics can be described well using a quantifiable
analytic perturbation until quite late in its evolution -
late enough for our argument (as shown in Sec. 4.4).
The approach presented here, using basic physics, is
intended as a pedagogical introduction to the physics of
gravitational wave signals, and as a tool to build intuition
using rough, but straightforward, checks. Our presenta-
tion here is by design elementary, but gives results con-
sistent with more advanced treatments. The fully rigor-
ous arguments, as well as precise numbers describing the
system, have already been published elsewhere [2–6].
The paper is organized as follows: our presentation
begins with the data output by the detectors.
2
Section 2
describes the properties of the signal read off the strain
data, and how they determine the quantities relevant for
analyzing the system as a binary inspiral. We then dis-
cuss in Sec. 3, using the simplest assumptions, how the
binary constituents must be heavy and small, consistent
only with being black holes. In Sec. 4 we examine and
justify the assumptions made, and constrain both
masses to be well above the heaviest known neutron
stars. Section 5 uses the peak gravitational wave lumi-
nosity to estimate the distance to the source, and calcu-
lates the total luminosity of the system. The appendices
provide a calculation of gravitational radiation strain and
radiated power (App. A), and discuss astrophysical com-
pact objects of high mass (App. B) as well as what one
might learn from the waveform after the peak (App. C).
2 Analyzing the observed data
Our starting point is shown in Fig. 1: the instrumentally
observed strain data
h
(
t
), after applying a band-pass
filter to the LIGO sensitive frequency band (35–350 Hz),
and a band-reject filter around known instrumental
noise frequencies [14]. The time-frequency behavior of
the signal is depicted in Fig. 2. An approximate version
of the time-frequency evolution can also be obtained
directly from the strain data in Fig. 1 by measuring the
time differences

t
between successive zero-crossings
3
2
The advanced LIGO detectors use laser interferometry to measure
the strain caused by passing gravitational waves.For details of how
the detectors work,see [1] and its references.
3
To resolve the crossing at
t
0
.
35
s,when the signal amplitude is
lower and the true waveform’s sign transitions are difficult to pin-
Figure 1
The instrumental strain data in the Livingston detector
(blue) and Hanford detector (red), as shown in Figure 1 of [1]. Both
have been bandpass- and notch-filtered. The Hanford strain has
been shifted back in time by 6.9 ms and inverted. Times shown are
relative to 09:50:45 Coordinated Universal Time (UTC) on Septem-
ber 14, 2015.
Figure 2
A representation of the strain-data as a time-frequency
plot (taken from [1]), where the increase in signal frequency
(“chirp”) can be traced over time.
and estimating
f
GW
=
1
/
(2

t
), without assuming a
waveform model. We plot the
8
/
3 power of these
estimated frequencies in Fig. 3, and explain its physical
relevance below.
The signal is dominated by several cycles of a wave
pattern whose amplitude is initially increasing, starting
from around the time mark 0.30 s. In this region the grav-
itational wave period is decreasing, thus the frequency
is increasing. After a time around 0.42 s, the amplitude
drops rapidly, and the frequency appears to stabilize.
The last clearly visible cycles (in both detectors, after ac-
counting for a 6.9 ms time-of-flight-delay [1]) indicate
that the final instantaneous frequency is above 200 Hz.
The entire visible part of the signal lasts for around 0
.
15s.
In general relativity, gravitational waves are produced
by accelerating masses [15]. Since the waveform clearly
shows at least eight oscillations, we know that a mass
point,we averaged the positions of the five adjacent zero-crossings
(over
6
ms).
(2 of 17) 1600209
C

2016 The Authors.
Annalen der Physik
published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim
www.ann-phys.org
Original Paper
Ann. Phys. (Berlin)
529
, No. 1–2 (2017)
Figure 3
A linear fit (green) of
f
8
/
3
GW
(
t
)
. While this interpolation
used the combined strain data from H1 and L1 (in fact, the sum
of L1 with time shifted and sign-flipped H1, as explained). A simi-
lar fit can be done using either H1 or L1 strain independently. The
fit shown has residual sum of squares
R
2
L1
H1
0
.
9
;wehave
also found
R
2
H1
0
.
9
and
R
2
L1
0
.
8
. The slope of this fitted line
gives an estimate of the chirp mass of
37 M

using Eq. 8. The
blue and red lines indicate
M
of 30
M

and 40
M

, respectively.
The error-bars have been estimated by repeating the procedure for
waves of the same amplitudes and frequencies added to the LIGO
strain data just before GW150914. A similar error estimate has been
found using the differences between H1 and L1 zero-crossings.
or masses are oscillating. The increase in gravitational
wave frequency and amplitude also indicate that during
this time the oscillation frequency of the source system
is increasing. This initial behavior cannot be due to a
perturbed system returning back to stable equilibrium,
since oscillations around equilibrium are generically
characterized by roughly constant frequencies and
decaying amplitudes. For example, in the case of a fluid
ball, the oscillations would be damped by viscous forces.
Here, the data demonstrate very different behavior.
During the period when the gravitational wave fre-
quency and amplitude are increasing, orbital motion of
two bodies is the only plausible explanation: there, the
only “damping forces” are provided by gravitational wave
emission, which brings the orbiting bodies closer (an “in-
spiral”), increasing the orbital frequency and amplifying
the gravitational wave energy output from the system.
4
Gravitational radiation has many aspects analogous
to electromagnetic (EM) radiation from accelerating
charges. A significant difference is that there is no analog
to EM dipole radiation, whose amplitude is proportional
4
The possibility of a different inspiraling system,whose evolution
is not governed by gravitational waves,is explored in App.A.1 and
shown to be inconsistent with this data.
to the second time derivative of the electric dipole mo-
ment. This is because the gravitational analog is the mass
dipole moment (

A
m
A
x
A
at leading order in the veloc-
ity) whose first time derivative is the total linear momen-
tum, which is conserved for a closed system, and whose
second derivative therefore vanishes. Hence, at leading
order, gravitational radiation is quadrupolar. Because the
quadrupole moment (defined in App. A) is symmetric
under rotations by
π
about the orbital axis, the radiation
has a frequency
twice
that of the orbital frequency (for a
detailed calculation for a 2-body system, see App. A and
pp. 356-357 of [16]).
The eight gravitational wave cycles of increasing fre-
quency therefore require at least four orbital revolutions,
at separations large enough (compared to the size of the
bodies) that the bodies do not collide. The rising fre-
quency signal eventually terminates, suggesting the end
of inspiraling orbital motion. As the amplitude decreases
and the frequency stabilizes the system returns to a sta-
ble equilibrium configuration. We shall show that the
only reasonable explanation for the observed frequency
evolution is that the system consisted of two black holes
that orbited each other and subsequently merged.
Determining the frequency at maximum strain am-
plitude
f
GW


max
: The single most important quantity for
the reasoning in this paper is the gravitational wave fre-
quency at which the waveform has maximum ampli-
tude. Using the zero-crossings around the peak of Fig. 1
and/or the brightest point of Fig. 2, we take the conser-
vative (low) value
f
GW


max
150 Hz
,
(2)
where here and elsewhere the notation indicates that the
quantity before the vertical line is evaluated at the time
indicated after the line. We thus interpret the observa-
tional data as indicating that the bodies were orbiting
each other (roughly Keplerian dynamics) up to at least
an orbital angular frequency
ω
Kep


max
=
2
π
f
GW


max
2
=
2
π
×
75 Hz
.
(3)
Determining the mass scale
: Einstein found [17] that
the gravitational wave strain
h
at a (luminosity) distance
d
L
from a system whose traceless mass quadrupole mo-
ment is
Q
ij
(defined in App. A) is
h
ij
=
2
G
c
4
d
L
d
2
Q
ij
d
t
2
,
(4)
and that the rate at which energy is carried away by
these gravitational waves is given by the quadrupole
C

2016 The Authors.
Annalen der Physik
published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim
(3 of 17) 1600209
www.ann-phys.org
Original Paper
B. P. Abbot et al.: The basic physics of the binary black hole merger GW150914
formula [17]
d
E
GW
d
t
=
c
3
16
π
G



̇
h


2
d
S
=
1
5
G
c
5
3

i
,
j
=
1
d
3
Q
ij
d
t
3
d
3
Q
ij
d
t
3
,
(5)
where


̇
h


2
=
3

i
,
j
=
1
d
h
ij
d
t
d
h
ij
d
t
,
theintegralisoverasphereatradius
d
L
(contributing
a factor 4
π
d
2
L
), and the quantity on the right-hand side
must be averaged over (say) one orbit.
5
In our case, Eq. 5 gives the rate of loss of orbital energy
to gravitational waves, when the velocities of the orbit-
ing objects are not too close to the speed of light, and the
strain is not too large [15]; we will apply it until the fre-
quency
f
GW


max
, see Sec. 4.4. This wave description is ap-
plicable in the “wave zone” [19], where the gravitational
field is weak and the expansion of the universe is ignored
(see Sec. 4.6).
For the binary system we denote the two masses by
m
1
and
m
2
, the total mass by
M
=
m
1
+
m
2
,andthere-
duced mass by
μ
=
m
1
m
2
/
M
. We define the mass ratio
q
=
m
1
/
m
2
and without loss of generality assume that
m
1
m
2
so that
q
1. To describe the gravitational wave
emission from a binary system, a useful mass quantity is
the
chirp mass
,
M
, related to the component masses by
M
=
(
m
1
m
2
)
3
/
5
(
m
1
+
m
2
)
1
/
5
.
(6)
Using Newton’s laws of motion, Newton’s universal
law of gravitation, and Einstein’s quadrupole formula for
the gravitational wave luminosity of a system, a simple
formula is derived in App. A (following [20, 21]) relating
the frequency and frequency derivative of emitted gravi-
tational waves to the chirp mass,
M
=
c
3
G


5
96

3
π
8
(
f
GW
)
11
̇
f
GW
3
1
/
5
,
(7)
where
̇
f
GW
=
d
f
GW
/
d
t
is the rate-of-change of the fre-
quency (see Eq. A5 and Eq. 3 of [22]). This equation is
expected to hold as long as the Newtonian approxima-
tion is valid (see Sec. 4.4).
Thus, a value for the chirp mass can be determined
directly from the observational data, using the frequency
and frequency derivative of the gravitational waves at
5
See App.A for a worked-out calculation,and pp.974-977 of [18] for
a derivation of these results,obtained by linearizing the Einstein
Equation,the central equation of general relativity.
any moment in time. For example, values for the fre-
quency can be estimated from the time-frequency plot
of the observed gravitational wave strain data (Fig. 2),
and for the frequency derivative by drawing tangents to
the same curve (see figure on journal cover). The time
interval during which the inspiral signal is in the sen-
sitive band of the detector (and hence is visible) corre-
sponds to gravitational wave frequencies in the range
30
<
f
GW
<
150 Hz. Over this time, the frequency (pe-
riod) varies by a factor of 5 (
1
5
), and the frequency deriva-
tive varies by more than two orders-of-magnitude. The
implied chirp mass value, however, remains constant to
within 35%. The exact value of
M
is not critical to the ar-
guments that we present here, so for simplicity we take
M
=
30 M

.
Note that the characteristic mass scale of the radiat-
ing system is obtained by direct inspection of the time-
frequency behavior of the observational data.
The fact that the chirp mass remains approximately
constant for
f
GW
<
150Hz is strong support for the orbital
interpretation. The fact that the amplitude of the grav-
itational wave strain increases with frequency also sup-
ports this interpretation, and suggests that the assump-
tions that go into the calculation which leads to these for-
mulae are applicable: the velocities in the binary system
are not too close to the speed of light, and the orbital mo-
tion has an adiabatically changing radius and a period
described instantaneously by Kepler’s laws. The data also
indicate that these assumptions certainly break down at
a gravitational wave frequency above
f
GW


max
, as the am-
plitude stops growing.
Alternatively, Eq. 7 can be integrated to obtain
f
8
/
3
GW
(
t
)
=
(8
π
)
8
/
3
5

G
M
c
3

5
/
3
(
t
c
t
)
,
(8)
which does not involve
̇
f
GW
explicitly, and can there-
fore be used to calculate
M
directly from the time peri-
ods between zero-crossings in the strain data. The con-
stant of integration
t
c
is the time of coalescense. We
have performed such an analysis, presented in Fig. 3,
to find similar results. We henceforth adopt a conserva-
tive lower estimate of
M
=
30 M

for the chirp mass. We
remark that this mass is derived from quantities mea-
sured in the detector frame, thus it and the quantities
we derive from it are given in the detector frame. Dis-
cussion of redshift from the source frame appears in
Sec. 4.6.
(4 of 17) 1600209
C

2016 The Authors.
Annalen der Physik
published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim
www.ann-phys.org
Original Paper
Ann. Phys. (Berlin)
529
, No. 1–2 (2017)
3 Evidence for compactness in the simplest
case
For simplicity, suppose that the two bodies have equal
masses,
m
1
=
m
2
. The value of the chirp mass then im-
plies that
m
1
=
m
2
=
2
1
/
5
M
=
35 M

, so that the total
mass would be
M
=
m
1
+
m
2
=
70 M

. We also assume
for now that the objects are not spinning, and that their
orbits remain Keplerian and essentially circular until the
point of peak amplitude.
Around the time of peak amplitude the bodies there-
fore had an orbital separation
R
given by
R
=

GM
ω
2
Kep


max
1
/
3
=
350 km
.
(9)
Compared to normal length scales for stars, this is a
tiny
value. This constrains the objects to be exceedingly
small, or else they would have collided and merged long
before reaching such close proximity. Main-sequence
stars have radii measured in hundreds of thousands or
millions of kilometers, and white dwarf (WD) stars have
radii which are typically ten thousand kilometers. Scal-
ing Eq. 9 shows that such stars’ inspiral evolution would
have terminated with a collision at an orbital frequency
of a few mHz (far below 1 Hz).
The most compact stars known are neutron stars,
which have radii of about ten kilometers. Two neutron
stars could have orbited at this separation without collid-
ing or merging together – but the maximum mass that a
neutron star can have before collapsing into a black hole
is about 3 M

(see App. B).
In our case, the bodies of mass
m
1
=
m
2
=
35 M

each have a Schwarzschild radius of 103 km. This is
illustrated in Fig. 4. The orbital separation of these
objects, 350 km, is only about twice the sum of their
Schwarzschild radii.
In order to quantify the closeness of the two objects
relative to their natural gravitational radius, we intro-
duce the compactness ratio
R
.Thisisdefinedasthe
Newtonian orbital separation between the centers of
the objects divided by the sum of their smallest possi-
ble respective radii (as compact objects). For the non-
spinning, circular orbit, equal-mass case just discussed
R
=
350 km
/
206 km
1
.
7.
For comparison with other known Keplerian systems,
the orbit of Mercury, the innermost planet in our solar
system, has
R
2
×
10
7
, the binary orbit for the stellar
Figure 4
A demonstration of the scale of the orbit at minimal
separation (black, 350 km) vs. the scale of the compact radii:
Schwarzschild (red, diameter 200 km) and extremal Kerr (blue, di-
ameter 100 km). Note the masses here are equal; as Sec. 4.2 ex-
plains, the system is even more compact for unequal masses.
While identification of a rigid reference frame for measuring dis-
tances between points is not unique in relativity, this complica-
tion only really arises with strong gravitational fields, while in the
Keplerian regime (of low compactness and low gravitational po-
tentials) the system’s center-of-mass rest-frame can be used.
Therefore if the system is claimed to be non-compact, the Keple-
rian argument should hold, and constrain the distances to be com-
pact. Thus the possibility of non-compactness is inconsistent with
the data; see also Sec. 4.4.
black hole in Cyg X-1
6
has
R
3
×
10
5
, and the binary
system of highest known orbital frequency, the WD sys-
tem HM Cancri (RX J0806), has
R
2
×
10
4
[24]. Obser-
vations of orbits around our galactic center indicate the
presence of a supermassive black hole, named Sgr A* [25,
26], with the star S2 orbiting it as close as
R
10
3
.Fora
system of two neutron stars just touching,
R
would be
between
2and
5.
The fact that the Newtonian/Keplerian evolution of
the orbit inferred from the signal of GW150914 breaks
down when the separation is about the order of the black
hole radii (compactness ratio
R
of order 1) is further evi-
dence that the objects are highly compact.
6
Radio,optical and X-ray telescopes have probed the accretion disk
extending much further inside [23].
C
©
2016 The Authors.
Annalen der Physik
published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim
(5 of 17) 1600209
www.ann-phys.org
Original Paper
B. P. Abbot et al.: The basic physics of the binary black hole merger GW150914
4 Revisiting the assumptions
In Sec. 3 we used the data to show that the coalescing
objects are black holes under the assumptions of a cir-
cular orbit, equal masses, and no spin. It is not possi-
ble, working at the level of approximation that we are
using here, to directly constrain these parameters of the
system (although more advanced techniques are able
to constrain them, see [2]). However, it is possible to
examine how these assumptions affect our conclusions
and in this section we show that relaxing them does not
significantly change the outcome. We also use the Keple-
rian approximation to discuss these three modifications
(Sec. 4.1–4.3), then revisit the Keplerian assumption it-
self, and discuss the consequences of foregoing it (Sec.
4.4–4.5). In Sec. 4.6 we discuss the distance to the source,
and its potential effects.
4.1 Orbital eccentricity
For non-circular orbits with eccentricity
e
>
0, the
R
of
Kepler’s third law (Eq. 9) no longer refers to the orbital
separation but rather to the semi-major axis. The instan-
taneous orbital separation
r
sep
is bounded from above by
R
, and from below by the point of closest approach (pe-
riapsis),
r
sep
1
e
R
. We thus see that the compact-
ness bound imposed by eccentric orbits is even tighter
(the compactness ratio
R
is smaller).
There is also a correction to the luminosity which
depends on the eccentricity. However, this correction is
significant only for highly eccentric orbits.
7
For these,
the signal should display a modulation [27]: the velocity
would be greater near periapsis than near apoapsis, so
the signal would alternate between high-amplitude and
low-amplitude peaks. Such modulation is not seen in the
data, whose amplitude grows monotonically.
This is not surprising, as the angular momentum that
gravitational waves carry away causes the orbits to circu-
larize much faster than they shrink [20, 21]. This correc-
tion can thus be neglected.
7
Eccentricity increases the luminosity [20,21] by a factor

(
e
)
=
1
e
2
7
/
2
1
+
73
24
e
2
+
37
96
e
4
1
,thus reducing the chirp
mass (inferred using Eq.7) to
M
(
e
)
=

3
/
5
(
e
)
·
M
(
e
=
0)
.Taking
into account the ratio between the separation at periapsis and the
semi-major axis,one obtains
R
(
e
)
=
1
e

2
/
5
(
e
)
·
R
(
e
=
0)
.
Hence for the compactness ratio to increase,the eccentricity must
be
e

0
.
6
,and for a factor of 2,
e

0
.
9
(see Fig. 5)
4.2 The case of unequal masses
It is easy to see that the compactness ratio
R
also gets
smaller with increasing mass-ratio, as that implies a
higher total mass for the observed value of the New-
tonian order chirp mass. To see this explicitly, we ex-
press the component masses and total mass in terms of
the chirp mass
M
and the mass ratio
q
,as
m
1
=
M
(1
+
q
)
1
/
5
q
2
/
5
,
m
2
=
M
(1
+
q
)
1
/
5
q
3
/
5
,and
M
=
m
1
+
m
2
=
M
(1
+
q
)
6
/
5
q
3
/
5
.
(10)
The compactness ratio
R
is the ratio of the orbital
separation
R
to the sum of the Schwarzschild radii of
the two component masses,
r
Schwarz
(
M
)
=
r
Schwarz
(
m
1
)
+
r
Schwarz
(
m
2
), giving
R
=
R
r
Schwarz
(
M
)
=
c
2
2(
ω
Kep


max
GM
)
2
/
3
=
c
2
2(
π
f
GW


max
G
M
)
2
/
3
q
2
/
5
(1
+
q
)
4
/
5
3
.
0
q
2
/
5
(1
+
q
)
4
/
5
.
(11)
This quantity is plotted in Fig. 5, which clearly shows that
for mass ratios
q
>
1 the compactness ratio
decreases
:the
separation between the objects becomes smaller when
measured in units of the sum of their Schwarzschild
radii. Thus, for a given chirp mass and orbital frequency,
a system composed of unequal masses is
more
compact
than one composed of equal masses.
One can also place an
upper
limit on the mass ratio
q
, thus a lower bound on the smaller mass
m
2
, based
purely on the data. This bound arises from minimal
compactness: we see from the compactness ratio plot
in Fig. 5 that beyond the mass ratio of
q
13 the
system becomes so compact that it will be within the
Schwarzschild radii of the combined mass of the two
bodies. This gives us a limit for the mass of the smaller
object
m
2
11 M

.Asthisis3–4timesmoremassive
than the neutron star limit, both bodies are expected to
be black holes .
4.3 The effect of objects’ spins
The third assumption we relax concerns the spins of the
objects. For a mass
m
with spin angular momentum
S
we
define the dimensionless spin parameter
χ
=
c
G
S
m
2
.
(12)
(6 of 17) 1600209
C

2016 The Authors.
Annalen der Physik
published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim
www.ann-phys.org
Original Paper
Ann. Phys. (Berlin)
529
, No. 1–2 (2017)
Figure 5
This figure shows the compactness ratio constraints im-
posed on the binary system by
M
=
30 M

and
f
GW


max
=
150 Hz
. It plots the compactness ratio (the ratio of the separa-
tion between the two objects to the sum of their Schwarzschild
radii) as a function of mass ratio and eccentricity from
e
=
0
to
the very high (arbitrary) value of
e
=
0
.
8
. The bottom-left corner
(
q
=
1
,
e
=
0
) corresponds to the case given in Sec. 3. At fixed
mass ratio, the system becomes more compact with growing ec-
centricity until
e
=
0
.
27
, as explained in Sec. 4.1. The bottom edge
(
e
=
0
)illustratestheargumentgiveninSec.4.2andEq.11:thesys-
tem becomes more compact as the mass ratio increases. We note
that (for
e
=
0
) beyond mass ratio of
q
13
(
m
2
11 M

)the
system would become more compact than the sum of the compo-
nent Schwarzschild radii.
The spins of
m
1
and
m
2
modify their gravitational radii
as described in this subsection, as well as the orbital dy-
namics, as described in the next subsection.
The smallest radius a non-spinning object (
χ
=
0) could have without being a black hole is its
Schwarzschild radius. Allowing the objects to have angu-
lar momentum (spin) pushes the limit down by a factor
of two, to the radius of an extremal Kerr black hole (for
which
χ
=
1),
r
EK
(
m
)
=
1
2
r
Schwarz
(
m
)
=
Gm
/
c
2
.Asthisis
linear in the mass, and summing radii linearly, we obtain
a lower limit on the Newtonian separation of two adja-
cent non-black hole bodies of total mass
M
is
r
EK
(
m
1
)
+
r
EK
(
m
2
)
=
1
2
r
Schwarz
(
M
)
=
GM
c
2
1
.
5

M
M


km
.
(13)
The compactness ratio can also be defined in relation to
r
EK
rather than
r
Schwarz
, which is at most a factor of two
larger than for non-spinning objects.
We may thus constrain the orbital compactness ratio
(now accounting for eccentricity, unequal masses, and
spin) by
R
=
r
sep
(
M
)
r
EK
(
M
)
R
(
M
)
r
EK
(
M
)
=
c
2
GM
ω
Kep
2
/
3
c
2
2
6
/
5
G
M
ω
Kep
2
/
3
=
c
2
2
6
/
5
π
G
M
f
GW


max
2
/
3
3
.
4
,
(14)
where in the last step we used
M
=
30 M

and
f
GW


max
=
150 Hz. This constrains the constituents to be under
3.4 (1.7) times their extremal Kerr (Schwarzschild) radii,
making them highly compact. The compact arrange-
ment is illustrated in Fig. 4.
We can also derive an upper limit on the value of the
mass ratio
q
, from the constraint that the compactness
ratio must be larger than unity. This is because, for a fixed
value of the chirp mass
M
and a fixed value of
f
GW


max
,
the compactness ratio
R
decreases as the mass ratio
q
increases. Thus, the constraint
R
1, puts a limit on the
maximal possible
q
and thus on the maximum total mass
M
max
,

M
max
M

3
.
4
3
/
2
×
2
6
/
5
14
.
4
,
(15)
which for GW150914 implies
M
max
432 M

(and
q
83). This again forces the smaller mass to be at least 5 M

– well above the neutron star mass limit (App. B).
The conclusion is the same as in the equal-mass or
non-spinning case: both objects must be black holes.
4.4 Newtonian dynamics and compactness
We now examine the applicability of Newtonian dynam-
ics. The dynamics will depart from the Newtonian ap-
proximation when the relative velocity
v
approaches
the speed of light or when the gravitational energy be-
comes large compared to the rest mass energy. For a bi-
nary system bound by gravity and with orbital velocity
v
, these two limits coincide and may be quantified by
the post-Newtonian (PN) parameter [28]
x
=
(
v
/
c
)
2
=
GM
/
c
2
r
sep
. Corrections to Newtonian dynamics may
be expanded in powers of
x
, and are enumerated by their
PN order. The 0PN approximation is precisely correct at
x
=
0, where dynamics are Newtonian and gravitational
wave emission is described exactly by the quadrupole
formula (Eq. 5).
C

2016 The Authors.
Annalen der Physik
published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim
(7 of 17) 1600209
www.ann-phys.org