# Does the observed period of a pulsar change with the time of year?

Over in the Physics SE a question was posted asking about the difference in the time dilation of the Earth between perihelion and aphelion:

Does Earth experience any significant, measurable time dilation at perihelion?

Rather to my surprise it turns out that because of the changes in the Earth-Sun distance and the Earth's orbital velocity there is a difference of about $60\mu$s per day between the two extremes.

A commentator pointed out that pulsars can be measured accurately enough to detect this difference. However I have never heard of a pulsar measurement having to be corrected for the time of year, and Googling has found me nothing related. I would be interested to know if this is something that needs to be considered.

The difference is slightly over one part in $10^9$, so presumably it depends on whether pulsars can be timed this accurately.

• This link discusses timekeeping accuracy in the comments and someone quotes $10^{-7}$ sec as an accuracy, which is too large for your $10^{-9}$ requirement, assuming a period of the order of $1$ sec. I don't think this is good enough to be an answer, but you may be able to make some use of it. Mar 29, 2017 at 12:09
• This is 100% a guess, but my assumption is that, since pulsars are so precise, their timing is calculated using GPS time, which I'm assuming already accounts for these types of time dilations (amongst others). Mar 29, 2017 at 15:32
• @zephyr -- GPS time does not account for these time dilations. GPS time is a fixed offset from International Atomic Time (TAI), which measures time at sea level on the surface of the Earth. The question at physics.SE that motivated this question that essentially asked about Barycentric Dynamical Time (TDB). Mar 29, 2017 at 15:42
• Possibly this paper (Edwards et al. 2006) on the pulsar timing code tempo2 (especially Section 2.1.5: "Einstein delay") might be useful: adsabs.harvard.edu/doi/10.1111/j.1365-2966.2006.10870.x Mar 30, 2017 at 11:15

Yes. In terms of pulsar timing measurements, this is a massive effect! A +/- 30 km/s doppler shift changes the pulsar frequency by +/- 1 part in 10000. This sounds small, but the accumulated phase shift over many periods is readily apparent. In addition, the light travel time across the solar system has to be taken into account, as well as the Earth's rotation and some other smaller effects - such as Shapiro delay.

If the question refers to the specific annually varying difference in clock rates caused by the different gravitational potential experienced by an Earthbound telescope on an elliptical (as opposed to circular) orbit -the answer is still yes.

This is item 4 in the list of applied corrections given on p.52 of "Pulsar Astronomy" by Lyne et al. The maximum effect is a rate change of $3\times 10^{-9}$ leading to a maximum lead or lag of 1.7 ms.

• I think the gist of the question was more about whether the difference in GR time dilation due to the changes in the Earth's distance from the Sun factors into pulsar timing calculations -- in addition to the Doppler shift and the Roehmer effect you mentioned. Mar 30, 2017 at 21:36

Longer answer is: correcting for the time dilation effects of Earth moving around the Sun's gravitational potential is actually relatively standard in almost all branches of astronomy. To the point where running that correction is a sentence in a paper (sometimes less), and is probably why you had trouble Googling for it.

(I'll caveat all this by saying I'm mostly familiar with exoplanet transit and RV timing issues, but they should be the same as what the pulsar folks have to deal with).

As some background, the base time-keeping system used around the world is International Atomic Time (TAI), which is a weighted average of over 300 atomic clocks determined by the International Bureau of Weights and Measures outside of Paris. Importantly, TAI is strictly continuous: there are no leap seconds added. This is important if you care about sub-second timing precision.

What we use as normal "clock" time is Coordinated Universal Time (UTC), which is TAI with leap seconds subtracted off. Those leap seconds are present to deal with the fact that 86,400 SI seconds are 1 to 3 milliseconds less than one mean Solar day, and so ensure that our clock time is linked to the position of the Sun. The most recent leap second was added just this past New Year's, making UTC = TAI - 37 seconds.

Even further down the time-keeping rabbit hole is Barycentric Dynamic Time (TDB), which accounts for the variable relativistic time dilation over the course of a year that you asked about. TDB has a fixed offset from TAI of 32.184 seconds due to how the zero-points of the the two systems were defined, and otherwise stays within 1.6 milliseconds of TAI - depending upon where Earth is in its orbit.

Effectively all precise times reported by astronomers these days are the barycentric Julian date in the Barycentric Dynamic Time system (BJD_TDB). This is the Julian date an event would appear to happen for an observer located at the Solar System's barycenter using TDB as their timekeeping system. Note that the fact this is at the SS barycenter does matter, since observations on Earth will see similar events up to ~16 minutes apart over the course of the year due to light-travel time delay (Roemer Delay, for the aficionados) across the Earth's orbit.

So yeah, this all has to be accounted for all the time. As I said, these days the transformation is standard enough that you usually just list a time as "BJD_TDB" and don't have to explicitly discuss the transformation.

PS - In case you're wondering why Barycentric Dynamic Time is abbreviated TDB and Coordinated Universal Time is UTC, it's because we all use the French abbreviations.

Yes, the frequency change $$\Delta f$$ is easily measurable and can be explained by the Doppler effect. Suppose the pulsar is in the plane of the ecliptic. Then the earth approaches at a certain point in time with the maximum speed $$v_{orbit}=30$$ km/s. Six months later it is moving away at -30 km/s. Since the EM waves propagate at the speed of light, the following applies:

$$\Delta f=f_{pulsar}\cdot {\left(\sqrt{\frac{c+v_{orbit}}{c-v_{orbit}}}-1 \right)} \approx f_{pulsar}\cdot\frac{v_{orbit}}{c} \approx \frac{f_{pulsar}}{10000}$$

The arrival times of the impulses change accordingly. If you know the precise direction to the pulsar, this effect can be hidden. The received signals are then converted as if the antenna were located at the center of gravity of the solar system.

This periodic frequency shift disappears for pulsars near the poles of the ecliptic.

It depends on the quantity of matter surrounding the pulsar. If the surrounding matter gets too close, via accretion, it will increase its rotational rate, making the pulse change.*

• I believe you have mis-read the question. Take care to read carefully and consider if the question already has good answers. In this case, the question is about variations in the apparent timing of pulsars as a result of the Earth's orbit. This question is also about 3 years old. You might try answering some new and unanswered questions. May 9, 2020 at 18:59