Binary Pulsar Timing & Gravitational Waves

Hello! Welcome to the next lesson in our journey through gravitational wave physics.

In our previous lesson, we accomplished a major step by applying the quadrupole formula to a binary star system. We derived the power it radiates in gravitational waves, $L_{G W}$ , and established the crucial physical consequence: the binary's orbit must decay. We found the instantaneous rate at which the orbital separation $l_{0}$ shrinks:

\frac{d l _{0}}{d t} = - \frac{64}{5} \frac{G ^{3}}{c ^{5}} \frac{μ M ^{2}}{l _{0}^{3}}

This result provided the theoretical underpinning for the first indirect detection of gravitational waves via the Hulse-Taylor binary pulsar.

Today, we will build directly on that foundation to achieve this lesson's objective: to estimate the orbital decay and gravitational wave signal from a binary system like the Hulse-Taylor pulsar. We will integrate the equation of motion to find how the orbit evolves over time and connect our results to two key types of observations:

The change in orbital period, which is precisely what was measured for the Hulse-Taylor system.
The "chirp" signal—the characteristic rising frequency and amplitude of gravitational waves from an inspiraling system—which is what detectors like LIGO and Virgo measure.

1. The Rate of Orbital Period Decay

For binary pulsars, the most precisely measured quantity is the orbital period, $P$ . Our first task is to translate the rate of orbital shrinkage, $\frac{d l _{0}}{d t}$ , into a rate of period change, $\frac{d P}{d t}$ . The key is the principle of energy conservation: the orbital energy lost by the binary is carried away by gravitational waves.

\frac{d E _{or b}}{d t} = - L_{G W}

From the previous lesson, we have the expressions for the orbital energy $E_{or b}$ and the radiated power $L_{G W}$ for a circular orbit:

$E_{or b} = - \frac{G μ M}{2 l _{0}}$
$L_{G W} = \frac{32}{5} \frac{G ^{4}}{c ^{5}} \frac{μ ^{2} M ^{3}}{l _{0}^{5}}$

To connect these to the period $P$ , we use Kepler's third law, which relates the period to the orbital separation: $P^{2} = \frac{4 π ^{2}}{GM} l_{0}^{3}$ .

The following resource provides a clear and direct derivation of $\frac{d P}{d t}$ .

THE QUADRUPOLE FORMALISM APPLIED TO BINARY ...

Let's follow the steps to derive the rate of change of the orbital period. This document carefully relates the change in orbital energy to the change in the period, arriving at a testable prediction.

Please study pages 29 and 30. Start from the heading 'Power radiated in gravitational waves by a binary system in circular orbit' and follow the derivation that begins with the energy conservation equation dE_orb/dt + L_GW = 0 (Eq. 22). Trace the steps through to Equation (26), which gives dP/dt in terms of L_GW and E_orb.

The key steps in the derivation are:

Express the time derivative of the orbital energy, $\frac{d E _{or b}}{d t}$ , in terms of the rate of change of the period, $\frac{d P}{d t}$ . This is done by first relating $E_{or b}$ to $l_{0}$ , and then $l_{0}$ to $P$ via Kepler's Law. This yields: $\frac{d E _{or b}}{d t} = - \frac{2}{3} \frac{E _{or b}}{P} \frac{d P}{d t}$
Substitute this into the energy conservation equation $\frac{d E _{or b}}{d t} = - L_{G W}$ : $- \frac{2}{3} \frac{E _{or b}}{P} \frac{d P}{d t} = - L_{G W}$
Solve for $\frac{d P}{d t}$ to get the final, elegant relation: $\frac{d P}{d t} = \frac{3}{2} \frac{P}{E _{or b}} L_{G W}$

Since $E_{or b}$ is negative, $\frac{d P}{d t}$ is also negative, confirming that the period decreases as the binary radiates energy.

Application to the Hulse-Taylor Pulsar (PSR B1913+16)

Now we can apply this result to the Hulse-Taylor system. The resource you just read does exactly this on page 30. Using the observed parameters for PSR B1913+16 (masses, period, etc.), it calculates the theoretical prediction for $\frac{d P}{d t}$ .

Observed Parameters:
- $m_{1} \approx 1.441 M_{⊙}$ , $m_{2} \approx 1.387 M_{⊙}$
- Orbital Period $P \approx 7.75$ hours $(27907 s)$
Prediction (for a circular orbit): $\frac{d P}{d t} \approx - 2.2 \times 1 0^{- 13} s/s$ .
Prediction (including eccentricity): The actual orbit is quite eccentric ( $e \approx 0.617$ ). A more complex calculation that accounts for this gives $\frac{d P}{d t} \approx - 2.4 \times 1 0^{- 12} s/s$ .
Observed Value: After decades of observation, the measured value is $\frac{d P}{d t}_{o b s} = - (2.4184 \pm 0.0009) \times 1 0^{- 12} s/s$ .

The agreement between theory and observation is astonishingly good, at the level of ~0.2%. This provided undeniable (though indirect) proof of the existence of gravitational waves and the validity of General Relativity's predictions for them.

Hulse-Taylor Pulsar: Observational Evidence for Gravitational Waves — The cumulative change in the orbital period of the Hulse-Taylor pulsar over 30 years. The points are the observational data. The solid line is the prediction from General Relativity based on the orbital decay we have just quantified. The remarkable agreement is a cornerstone of experimental gravity.

2. The Gravitational Wave "Chirp" Signal

While binary pulsars provide indirect evidence through orbital decay, detectors like LIGO and Virgo aim to observe the gravitational waves directly. What does the signal from an inspiraling binary look like? It's characterized by a "chirp"—an increase in both frequency and amplitude as the system approaches merger.

The Chirp Mass

To analyze the chirp, it's useful to derive the evolution of the gravitational wave frequency, $f_{G W}$ . The frequency of the wave is twice the orbital frequency ( $f_{G W} = 2 f_{or b} = ω_{K} / π$ ).

arXiv:2303.11713v1 [gr-qc] 21 Mar 2023

Let's now derive the evolution of the GW frequency and introduce a fundamentally important quantity: the chirp mass. This quantity alone determines the rate of frequency change at the leading order.

Please read the section 'VI. THE INSPIRAL AND MERGER OF BINARY SYSTEMS OF COMPACT OBJECTS', starting from Equation (187) to the end of the page (Equation 190). Follow the derivation that leads to the rate of change of the GW frequency, df/dt (Eq. 189), and pay close attention to the definition and significance of the chirp mass, M_c.

As shown in the resource, by starting with the rate of energy loss and relating it to the change in orbital frequency, we arrive at:

\frac{d f _{G W}}{d t} = \frac{96}{5 c ^{5}} π^{8/3} (G M_{c})^{5/3} f_{G W}^{11/3}

where $M_{c}$ is the chirp mass, defined as:

M_{c} = \frac{( m _{1} m _{2} ) ^{3/5}}{( m _{1} + m _{2} ) ^{1/5}} = μ^{3/5} M^{2/5}

This is a crucial result. The rate at which the gravitational wave frequency increases (the "chirp") depends on this specific combination of the binary's masses. This means that by measuring the evolution of a signal's frequency, we can directly measure the chirp mass of the source system, even if it's millions of light-years away.

Test your understanding!

Consider a binary system of two equal-mass black holes, $m_{1} = m_{2} = m$ . What is its chirp mass $M_{c}$ in terms of $m$ ? How does this compare to the total mass $M = 2 m$ ?

Show answer

For an equal-mass binary, the total mass is $M = 2 m$ and the reduced mass is $μ = \frac{m \cdot m}{m + m} = \frac{m ^{2}}{2 m} = \frac{m}{2}$ .

The chirp mass is:

M_{c} = μ^{3/5} M^{2/5} = (\frac{m}{2})^{3/5} (2 m)^{2/5} = \frac{m ^{3/5}}{2 ^{3/5}} 2^{2/5} m^{2/5} = \frac{m ^{(3/5 + 2/5)}}{2 ^{(3/5 - 2/5)}} = \frac{m}{2 ^{1/5}}

Since $2^{1/5} \approx 1.1487$ , we have $M_{c} \approx \frac{m}{1.1487} \approx 0.87 m$ .
The total mass is $M = 2 m$ , so the chirp mass is about 43.5% of the total mass for an equal-mass system.

Time to Coalescence

The equation for $\frac{d f _{G W}}{d t}$ also allows us to calculate the remaining time until the two bodies merge. By separating variables and integrating, we can find the time $τ$ it takes for the frequency to evolve from its current value $f$ to infinity (the point of coalescence).

THE QUADRUPOLE FORMALISM APPLIED TO BINARY ...

Finally, let's see how we can calculate the time remaining until merger and how the wave's amplitude and frequency evolve during the final inspiral.

Please study pages 33 and 34, from the heading 'WAVEFORM: AMPLITUDE AND PHASE' up to the end of the page. This section solves for the evolution of the frequency ν_GW(t) and the amplitude h_0(t), and introduces the integrated phase Φ(t). Focus on how ν_GW and h_0 depend on the time to coalescence.

As the reading shows, integrating the frequency evolution gives:

τ (f) = t_{co a l} - t = \frac{5}{256} (G M_{c})^{- 5/3} (π f)^{- 8/3} c^{5}

This formula tells us the remaining lifetime of a binary system, given its chirp mass and current gravitational wave frequency. For example, for a binary neutron star system with $M_{c} \approx 1.2 M_{⊙}$ emitting waves at 10 Hz, the remaining time to merger is about 5 minutes.

The frequency and amplitude evolve as a function of the time to coalescence, $τ = t_{co a l} - t$ :

Frequency: $f_{G W} (τ) \propto τ^{- 3/8}$
Amplitude: $h (τ) \propto f_{G W}^{2/3} \propto τ^{- 1/4}$

Both the frequency and amplitude diverge as $τ \to 0$ , producing the characteristic chirp that signals the final, violent moments of a binary's life.

Conclusion

In this lesson, we have taken the final step in applying the quadrupole formalism to its most important astrophysical source. We have translated the abstract concept of gravitational-wave energy loss into concrete, observable predictions that have been spectacularly confirmed.

Key Takeaways:

The energy loss due to gravitational waves causes the orbital period $P$ of a binary to decrease at a rate $\frac{d P}{d t} \propto \frac{μ M ^{2}}{l _{0}^{4}} P$ .
The comparison of this predicted rate with decades of observation of the Hulse-Taylor binary pulsar provided the first indirect but compelling evidence for gravitational waves.
The evolution of the gravitational wave frequency from an inspiraling binary is governed by the chirp mass, $M_{c}$ . Measuring the frequency "chirp" directly measures $M_{c}$ .
The frequency and amplitude of the gravitational wave signal increase as the binary inspirals, scaling with the time to coalescence $τ$ as $f \propto τ^{- 3/8}$ and $h \propto τ^{- 1/4}$ .

This completes our initial study of gravitational wave generation from binary systems. We have seen how to derive the wave properties and the backreaction on the source.

Preview of Next Lesson:

We will move on to applying our derived knowledge to the Hulse-Taylor binary pulsar system specifically. In the next lesson, we will estimate the orbital decay and gravitational wave signal from a binary system like the Hulse-Taylor pulsar, using the actual parameters of this system to calculate concrete numerical values for its orbital decay and expected signal. This will provide a capstone to our study of gravitational wave sources.

(Correction: The preview text from the thought process was slightly misaligned with the current lesson's title. The correct preview is below.)