Gravitational Radiation 1

One of the core themes of Physics from Planet Earth (PPE) is that many of the spectacular discoveries of contemporary astronomy, astrophysics and cosmology can be intelligently addressed in introductory physics classes.  Beginning students can appreciate and – at some level – comprehend the exciting developments that are profoundly enlarging our understanding of the universe.  But if such discoveries are to be included in the introductory curriculum, they must be tied tightly to the physics principles that form the bedrock of the first-year physics sequence.  Exposition must be kept simple, and “telling” must be severely limited.  The recent direct detection of gravitational waves offers a superb opportunity to engage students in contemporary astrophysics.  In this and the following post, we outline materials (suitable for lecture and homework) that might be used to introduce gravitational radiation to a first-year audience.  This post will discuss the  indirect observation of gravitational radiation (in 1975) by Taylor and Hulse (who shared the 1993 Nobel Prize in Physics), Weisberg and others.  The second post will deal with the recent (2016) direct observation of gravitational waves by the LIGO Scientific Collaboration, a global consortium of over 1000 scientists from 83 institutions, spearheaded by Weiss, Thorne, and Drever.  We hope these materials will encourage instructors to address this exciting topic in their introductory mechanics classes.

 

A.  Discovery of the Hulse-Taylor Binary Pulsar 1913+16

Pulsars were discovered by Joscelyn Bell Brunell and Anthony Hewish in 1967 (See PPE Chapter 11).  Just six years later, when graduate student Russell Hulse joined Joseph Taylor’s research group at the University of Massachusetts, several dozen pulsars had been discovered, and searches for new pulsar had already lost their novelty.  Nevertheless, Taylor believed a new survey with much higher sensitivity and better signal processing could provide a reliable estimate of the total number of pulsars in the galaxy, and clarify how pulsars fit into the overall process of stellar evolution.  Hulse was dispatched to the Arecibo Radio Observatory (the world’s largest radio telescope) in Puerto Rico to search for new pulsars, and by the summer of 1974 he had found 32 of them.  The most interesting was PSR 1913+16 (the numbers refer to its celestial coordinates: right ascension 19h 13m and declination 16°), whose ∼59 ms period was the second fastest pulsar period on record at that time.  While that in itself was remarkable, a vastly more important discovery awaited Hulse’s notice.

Recall that a pulsar’s period P (the time between two detected pulses) is equal to its period of rotation (PPE, Section 11.6).  Due to electromagnetic breaking, the pulsar spins down and P typically increases at a rate \large dP/dt<10^{^{-12}}, i.e., by < 10μs/yr.  This extremely high stability makes pulsars the equivalent of giant atomic clocks.  But Hulse found that PSR 1913+16’s period changed by as much as 8 μs within 5 minutes, and by as much as 80 μs in one day!  When he first observed the pulsar, he found that its period decreased throughout the 2 hour observing window allotted to him each day.  What’s more, he noticed that the values of P(t) taken the following day matched those measured 24 hours earlier if they were shifted by 45 minutes. In other words, P(t) was a periodic function of time!  Finding no evidence of instrumental error, he guessed that the pulsar had an undetected companion star, and that the two stars were in orbit about their common center of mass.  Similar to the Doppler effect, the measured spacing between pulses would shrink when the pulsar was approaching Earth, and would grow when the pulsar was receding from Earth.  If Hulse’s guess was right, PSR 1913+16 would be the first pulsar to be discovered in a binary system, and – at the very least – it would allow him to determine a pulsar mass unambiguously, something that had never been done before.  During the next two weeks, the daily downward drift in the period stopped, reversed itself, and became an upward drift, just as Hulse hoped.  Elated, he relayed his news to Taylor (by shortwave radio, since telephone communications between Puerto Rico and Massachusetts were unreliable), who immediately grasped its significance and hopped the next flight to San Juan.

  1.  PSR 1913+16 has an orbital period of 0.322997 d.  Explain why Hulse’s  measurements of P(t) were repeatable on the following day (i.e., 24 hours later) if the later data were shifted by 45 minutes.

 

B.  Gravitational Radiation Background

In 1916, Albert Einstein published his theory of General Relativity (GR), which radically changed our intuitive conceptions of fixed space and universal time.  Among the testable predictions of GR are (a) light is deflected by gravity (gravitational lensing), (b) time flows faster in regions of higher gravitational potential (gravitational red shift), and (c) accelerating masses emit gravitational waves that propagate at the speed of light.   The first two predictions have been extensively verified (see PPE Chapter 7, Box 7.1), but gravitational waves went undetected – and their existence was hotly disputed – until Taylor and Hulse’s discovery of PSR 1913+16.

For simplicity, consider a binary system of two equal-mass stars (\large m_{1}=m_{2}=M/2, where M is the total mass), each in a circular orbit of radius r about the system’s center of mass.  Circular motion is a superposition of oscillations along  x– and y-axes, and GR predicts that the system will lose energy by emitting gravitational waves.  For this simple case, the predicted gravitational wave luminosity is

\large L_{cir}=\frac{2}{5}\frac{G}{c^{5}}M^{^{2}}a^{4}\omega _{orb}^{6}

(1)

where \large a=2r is the separation between the stars, and \large \omega _{orb}=2\pi /T_{orb} is the angular velocity of either star in its orbit.  From Newton’s second law, \large m\omega _{orb}^{2}r=Gm^{2}/4r^{2}, or \large \omega _{orb}^{2}=Gm/4r^{3}=GM/a^{3}.  The total energy of the binary system is

\large E=K+U=2\left ( \frac{1}{2}m\omega_{orb}^{2}r^{2} \right )-Gm^{2}/2r = -\frac{Gm^{2}}{4r}=-\frac{GM^{2}}{8a},

and the orbital radius shrinks as the system loses energy:

\large \frac{\mathrm{d} E}{\mathrm{d} t}=\frac{GM^{2}}{8a^{2}}\frac{\mathrm{d} a}{\mathrm{d} t}=-L_{cir}

or

\large \frac{\mathrm{d} a}{\mathrm{d} t}=-\frac{16}{5}\frac{G^{3}M^{3}}{a^{3}c^{5}}.

Finally, since \large \omega _{orb}\propto a^{-3/2}, and the orbital period \large T=2\pi /\omega _{orb},

\large \frac{1}{T}\frac{\mathrm{d} T}{\mathrm{d} t}=-\frac{1}{\omega _{orb}}\frac{\mathrm{d} \omega _{orb}}{\mathrm{d} t}=\frac{3}{2}\frac{1}{a}\frac{\mathrm{d} a}{\mathrm{d} t}

or

\large \frac{1}{T}\frac{\mathrm{d} T}{\mathrm{d} t}=-\frac{24}{5}\frac{G^{3}M^{3}}{a^{4}c^{5}}.

(2)

Eqn. 2 expresses the fractional rate of change of the pulsar’s orbital period due to gravitational radiation, for a binary system of equal mass stars in circular orbit.

2.    Show that Eqn. 2 is dimensionally correct, i.e., that the RHS has the units \large s^{-1}.

3.   To appreciate the uniqueness of the Hulse-Taylor binary, imagine two Sun-like stars in circular orbit 1 AU apart.  (a) What is their orbital period in years?  (b) How close would the two stars need to be to have a period of 0.365 d (= .001 yr), which is comparable to PSR 1913+16?  (Hint: recall \large 4\pi^{2}/GM_{Sun}=1\: \textup{yr}^{2}/\textup{AU}^{3})  (Ans: (a) .707 yr  (b) .0126 AU  Compare this to Mercury’s semimajor axis of 0.39 AU.)

C.  Gravitational Waves and the Hulse-Taylor Binary Pulsar

By the end of September, 1974, Hulse and Taylor had obtained a complete radial velocity curve for PSR 1913+16 (Figure 1), and it revealed that the pulsar and its companion are in elliptical orbits with eccentricity ε = 0.617.  While they could not initially determine the inclination of the orbit (the plane of the orbit does not lie in the plane of Fig. 1), they knew that at periastron, when the two stars are

Grav Wave Fig. 1
Figure 1: (top) Radial velocity curve of pulsar measured by Hulse and Taylor; (bottom) Approximate orientation of the binary relative to Earth. The orbit is inclined by about 45 degrees relative to the plane of the figure.

closest to the CM and to each other, the pulsar’s speed is at least 300 km/s = .001c.  This suggested to Taylor that PSR 1913+16 could serve as an ideal proving ground for general relativity.  The full orbital details and the masses \LARGE m_{P} and \LARGE m_{C} of the two stars were determined using a combination of Kepler’s laws and general relativity, with these results:

\large m_{P}=1.42\, m_{Sun}\; \approx \; m_{C}=1.41\, m_{Sun}

\large a_{C}\simeq a_{P}=9.762\times 10^{8}\: \textup{m}

\large T=0.3230\, \textup{d}=2.791\times 10^{4}\: \textup{s}

where the subscripts P and C refer to the pulsar and its companion, and \large a_{P}\, (a_{C}) is the semimajor axis of the pulsar (companion) orbit.

4.  Show that these results are consistent with Kepler’s 3rd law (see PPE section 8.11).

Since the two stars are in elliptical orbit, the magnitude of their acceleration changes with time.  In fact, the acceleration at periastron is nearly 20 times greater than at apastron.  To calculate the time-averaged gravitational luminosity, Eqn. 1 must be multiplied by a function of the eccentricity:

\large f(\epsilon )=(1-\epsilon ^{2})^{-7/2}\left ( 1+\frac{73}{24}\epsilon ^{2}+\frac{37}{96}\epsilon ^{4} \right ) .

Finally, the expression for the luminosity of a binary with unequal masses (\large m_{P}\neq m_{C}) is

\large L=\frac{32}{5}\frac{G}{c^{5}}\, \mu ^{2}a^{4}\omega _{orb}^{6}\, f(\epsilon )

(3)

where \large a=a_{P}+a_{C} is the semi-major axis of the relative motion, \large \mu =m_{P}m_{C}/(m_{P}+m_{C}) is the reduced mass of the system, and \large \omega _{orb}^{2}=(2\pi /T)^{2}=GM/a^{3}, as usual.  For the Hulse-Taylor binary, \large m_{P}\simeq m_{C} so \large \mu \simeq M/4.

5.  Show that Eqn. 3 reduces to Eqn. 1 for two identical stars in circular orbit.

6.  Calculate the average power emitted by gravitational radiation from the Hulse-Taylor binary.  Compare your answer to the electromagnetic power radiated by the Sun.  (Ans: \large 7.4\times 10^{24}\, \, \textup{W}\approx 0.02\: L_{Sun})

7.  By what amount is \large a
decreasing each year?  (Ans: ∼ 3.5 m/yr)

Grav Wave Fig 2
Figure 2: Orbital decay of PSR 1913+16 as reported in Ref. 4. The dots are the data and the line is the prediction from GR.

8.  By how much does the orbital period decrease each year? (Ans: ∼ 80 μs/yr)

Taylor and colleagues continued to observe PSR 1913+16 over the next three decades, periodically reporting their measurements of the system’s orbital period.  Their results (as of 2010) are summarized in Figure 2, which is a plot of the accumulated advance of the periastron time relative to the time expected for an orbit that does not decay with time.  The points represent their data, and the line is the expected general relativistic result calculated using Eqn. 3.  Error bars are “mostly too small to see” in the figure.  As you can judge, the agreement with theory is spectacular.  Beyond a doubt, Taylor, Hulse, and Taylor’s later colleagues had demonstrated the existence of gravitational waves!

It seems pedagogically appropriate to explain the “inverted parabola” shape of the curve.  Students (as well as this writer) might naively expect the data to lie on a straight line.  But it is not the orbital period that is plotted in Fig. 2 (which would lie on a straight line).  Instead, it is the difference between the actual time when the pulsar arrives at periastron and the time it would have arrived if its orbit did not change.  Let \large T_{0} be the initially measured period (i.e., in 1975), and let \large T_{i} be the period of its ith full orbit after the initial measurement.  Let \Delta =\left | \frac{1}{T}\frac{\mathrm{d} T}{\mathrm{d} t} \right |\ll 1 be the fractional rate of change in period calculated using Eqn. 2.  Then,

\large T_{1}=T_{0}(1-\Delta )

\large T_{2}=T_{1}(1-\Delta )=T_{0}(1-\Delta )^{2}\simeq T_{0}(1-2\Delta )

\large \vdots

\large T_{i}=T_{0}(1-\Delta )^{i}\simeq T_{0}(1-i\Delta ).

After the nth orbit, the pulsar returns to periastron at time \large t_{n}=\sum_{i=1}^{n}T_{i} and the accumulated shift in arrival time is

T_{0}\sum_{i=1}^{n}(1-\Delta )^{i}-nT_{0}=-T_{0}\Delta \sum_{i=1}^{n}i

\large =-T_{0}\Delta\, \, \frac{n(n+1)}{2}.

The system completes about 3 orbits per day, or over 1100 orbits per year, so for the data displayed in Fig. 2, \large n\gg 1 and \large n(n+1)\simeq n^{2}.  The accumulated shift in periastron arrival time (the ordinate of Fig. 2) becomes \large -\Delta t^{2}/2T_{0}, i.e., the curve is parabolic.

9.  How many orbits did the binary system complete in the 30 year interval 1975-2005?  Calculate the accumulated shift in periastron time predicted by general relativity, and compare your answer to the data shown in Fig. 2.  (Ans:  \large n=3.39\times 10^{4}\large \Delta t=43.2 \: \textup{s})

References:

  1.  Russell A. Hulse Nobel Lecture, http://www.nobelprize.org/nobel_prizes/physics/laureates/1993/hulse-lecture.pdf
  2. R. A. Hulse and J. H. Taylor, Ap. J. 195: L51-L53 (1975)
  3. J. H. Taylor et al, Ap. J. 206: L53-L58 (1976)
  4. J. M. Weisberg, D. J. Nice, and J. H. Taylor, Ap. J. 722: 1030-1034 (2010)
  5. Kostas D. Kokkotas, “Gravitational Wave Physics,” http://www.tat.physik.uni-tuebingen.de/~kokkotas/Teaching/NS.BH.GW_files/GW_Physics.pdf
Advertisements