Double-Neutron-Star Binary PSR B1913+16

The double-neutron-star binary discussed in lectures contains a neutron star from which we detect a pulsed radio signal, named PSR B1913+16, and a second neutron star which we cannot see. The radio pulses arise because PSR B1913+16 emits a beam of radio waves along its magnetic axis, which is tilted with respect to its rotation axis, and the beam sweeps past the Earth once per neutron-star rotation, as in Fig. 2.1. (Just like a lighthouse.) The second neutron star probably emits a beam of radio waves too, but this second beam presumably never sweeps past the Earth.

 
Figure: 2.1

PSR B1913+16 rotates once every as measured by an observer in its orbital rest frame. However, the separation of the radio pulses measured at Earth (i.e. the pulse period) fluctuates between and , as shown in Fig. 2.2, mainly due to the Doppler effect. (This is related to, but not the same as, the Doppler shift of spectral lines discussed in lectures.)

 
Figure: 2.2

1. Redraw Fig. 2.2 and mark three points where PSR B1913+16 is moving (i) towards Earth with maximum velocity along the line of sight, (ii) away from Earth with maximum velocity along the line of sight, and (iii) transverse to the line of sight.
2. What property of the binary system corresponds to the time interval of ?

By studying the detailed shape of the curve in Fig. 2.2, including general relativistic effects, it is possible to deduce the eccentricity of the binary orbit, e=0.617, and the masses of the two neutron stars, and . ( is PSR B1913+16.)

3. Using your answer to part 2. and the data in the preceding paragraph, calculate the lengths of the semimajor and semiminor axes of the orbit of . Then do the same for .
4. Using your answer to part 3., draw approximately to scale the two stellar orbits, placing the centre of mass at the origin. Choose your scale wisely so that the two orbits fit on a page and can be seen without a magnifying glass!

In reality, the pulse-period fluctuations in Fig. 2.2 are known to great precision; the pulse period is measured to more than ten decimal places (!) using an atomic clock. However, the preceding parts of the question contain enough information for us to calculate ourselves.

Firstly, we need to calculate the maximum velocity of PSR B1913+16 along our line of sight. Assume for simplicity, here and in what follows, that the orbital plane is inclined edge-on to the line of sight and the semimajor axis is transverse to the line of sight. (In reality, this is not the case; the inclination angle i is known to be .) Note that we cannot just apply the results of Shu, Problem 10.1, because that problem pertains to circular orbits (e=0).

5. It was shown in lectures that the relative position of the two stars, , satisfies the 1-body Kepler equation of motion

   

   Write down without proof the energy equation corresponding to (1). (Adapt eq. 3 on p. 23 of the lecture notes.)

6. By expressing E in terms of the semimajor axis a (adapt the result from p. 25 of the lecture notes), show that the energy equation corresponding to (1) takes the form

   

7. Explain in words at what point in the orbit the velocity along the line of sight is a maximum. Substitute into (2) the value of at this point and show that one has

   

8. The previous result is the maximum velocity of the relative position vector . Using the results presented in lectures, relate the actual position of PSR B1913+16 to the relative position (with the centre of mass at the origin). Hence find an expression for the maximum value of the velocity of (i.e. PSR B1913+16).
9. Evaluate numerically the maximum value of the velocity of PSR B1913+16. Express your answer in .

Now that we know the maximum velocity of PSR B1913+16, we relate it to the Doppler shift of the pulse period, .

Consider a flashlight blinking periodically, with period in its rest frame. Suppose that, at t=0, it blinks on at a distance L from the Earth, moving towards the Earth with speed V.

10. Find an expression, in terms of L and c, for the time when an observer on Earth sees the first flash from the flashlight. Find an expression, in terms of L, V, and c, for the time when an observer on Earth sees the second flash from the flashlight. (A quick diagram will help you answer this question.)
11. Using the results of part 10., find an expression (in terms of V, and c) for the period of the blinking flashlight measured at Earth.
12. Using the results of parts 9. and 11., evaluate in Fig. 2.2 numerically. Express your answer in .