<p>I do not think there is a definitive list of post-Keplerian parameters.  I would argue there is some ambiguity in the definition of a post-Keplerian parameter.  Kramer et al. cite <a href="https://ui.adsabs.harvard.edu/abs/1986AIHS...44..263D/abstract" rel="noreferrer">Damour &amp; Deruelle (1986)</a> (<a href="https://www.ihes.fr/%7Edamour/Conferences/DamourDeruelleAIHP86.pdf" rel="noreferrer">direct PDF</a>) as the source of their post-Keplerian timing model.  I would call that a great source for GR effects in pulsar timing, but it doesn't use the &quot;post-Keplerian&quot; language.</p>
<p>The standard appears to be any parameter that defines a GR effect in the pulsar timing model is called post-Keplerian.</p>
<p>The six Keplerian orbital elements fully specify the shape of a two-body Keplerian orbit.  The two-body problem does not have an exact solution in GR.  Solving it requires a perturbative post-Newtonian expansion.  Depending on how far one takes the expansion there are more or fewer unique parameters to measure.</p>
<p>In pulsar timing the model being fit is for the arrival times of radio pulses, not the orbital motion.  One can infer the orbital motion from the pulse arrival times, but there's a bit more going on.  For instance if a radio pulse passes nearby the pulsar's binary companion it will experience Shapiro delay, changing its time of arrival.  This effect is related to the binary orbit, but doesn't strictly speaking define the orbit in the same way Keplerian orbital elements do.  The Shapiro delay is certainly a non-Newtonian effect on the timing model, and the parameters that define it are grouped with the the post-Keplerian parameters.</p>
<p>For instance section 4 of a Living Review by <a href="https://link.springer.com/article/10.12942/lrr-2003-5" rel="noreferrer">Stairs (2003)</a> defines five post-Keplerian parameters:</p>
<ul>
<li><span class="math-container">$\dot\omega$</span> - advance of periastron (<span class="math-container">$k$</span> is a different parameterization of the same thing)</li>
<li><span class="math-container">$\gamma_E$</span> - Einstein delay amplitude</li>
<li><span class="math-container">$\dot{P}_b$</span> - binary orbital period derivative</li>
<li><span class="math-container">$r$</span> - Shapiro delay &quot;range&quot; parameter</li>
<li><span class="math-container">$s$</span> - Shapiro delay &quot;shape&quot; parameter</li>
</ul>
<p>I would say that <span class="math-container">$\dot\omega$</span> and <span class="math-container">$\dot{P}_b$</span> clearly fit into the classification of orbital shape. The Shapiro and Einstein delay are GR effects that affect the pulse arrival times and depend on the masses and positions of the binary, but don't really define the orbital shape in the same way.</p>
<p>In addition to the usual five, <a href="https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041050" rel="noreferrer">Kramer et al. (2021)</a> measure:</p>
<ul>
<li><span class="math-container">$\Omega_B^\mathrm{spin}$</span> - rate of relativistic spin precession of pulsar B</li>
<li><span class="math-container">$\delta_\theta$</span> - relativistic deformation of the orbit</li>
</ul>
<p>The determination of <span class="math-container">$\Omega_B^\mathrm{spin}$</span> comes from the eclipsing of pulsar A by pulsar B.  The orientation of B's magnetosphere affects the eclipsing, which is how it is measured.  The spin precession contains a relativistic contribution, <span class="math-container">$\Omega_B^\mathrm{spin}$</span>, which is the non-Newtonian effect.  This particular measurement is pretty unique to the double pulsar (PSR J0737–3039A/B).  I don't think there is any other system where it could be measured.</p>
<p><span class="math-container">$\delta_\theta$</span> is related to the modification of the binary eccentricity due to GR affects.  There's a second GR eccentricity modification <span class="math-container">$\delta_r$</span>, which cannot be measured for this system due to covariance with the pulsar spin frequency derivatives.</p>
<p>Damour &amp; Deruelle (1986) state in section 3.7 that <span class="math-container">$\delta_\theta$</span> and <span class="math-container">$\delta_r$</span> are immeasurable due to covariances with the non-relativistic timing model, which probably explains why they are omitted from some references.  They do say:</p>
<blockquote>
<p>Note however that in the long term <span class="math-container">$\delta_\theta$</span> can in principle be measured when <span class="math-container">$\omega$</span> has changed by a sufficient amount, but previous experience with measurement of <span class="math-container">$\gamma_E$</span> suggests it will be very difficult to measure <span class="math-container">$\delta_\theta$</span> with any decent accuracy.</p>
</blockquote>
<p>I think that Kramer et al. (2021) is the first time <span class="math-container">$\delta_\theta$</span> has been measured, but I'm not 100% sure.</p>

CC BY-SA 4.0
2021-12-15T23:01:22.867Z
Section 2.7.5 of [Edwards, Hobbs and Manchester 2006](https://academic.oup.com/mnras/article/372/4/1549/1186764#20550482) on the `TEMPO2` code's timing model references all 7 of the Post-Keplerian parameters. It also used the Damour and Deruelle model and references similar papers by the same people as your answer. That section also present formula for calculating the values, under the assumption that GR is correct, which is one of the tests discussed in Kramer et al 2021.
astrosnapper
CC BY-SA 4.0
2021-12-14T17:46:41.140Z
2021-12-14T17:46:41.140Z
Paul T.
<p><a href="https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041050" rel="noreferrer">Strong-Field Gravity Tests with the Double Pulsar</a> is a pretty big deal! 16 years of precision timing including &quot;timing parallax&quot; via VLBI has provided a dataset for which analysis requires (and <em>tests</em>) several GR effects simultaneously.</p>
<p>Among everything else it provides, I think this paper can be <em>a rich source of Stack Exchange questions and answer.</em></p>
<blockquote>
<p>ABSTRACT</p>
<p>Continued timing observations of the double pulsar PSR J0737–3039A/B, which consists of two active radio pulsars (A and B) that orbit each other with a period of 2.45 h in a mildly eccentric (e=0.088) binary system, have led to large improvements in the measurement of relativistic effects in this system. With a 16-yr data span, the results enable precision tests of theories of gravity for strongly self-gravitating bodies and also reveal new relativistic effects that have been expected but are now observed for the first time. These include effects of light propagation in strong gravitational fields which are currently not testable by any other method. In particular, we observe the effects of retardation and aberrational light bending that allow determination of the spin direction of the pulsar. <strong>In total, we detect seven post-Keplerian parameters in this system, more than for any other known binary pulsar.</strong> For some of these effects, the measurement precision is now so high that for the first time we have to take higher-order contributions into account. These include the contribution of the A pulsar’s effective mass loss (due to spin-down) to the observed orbital period decay, a relativistic deformation of the orbit, and the effects of the equation of state of superdense matter on the observed post-Keplerian parameters via relativistic spin-orbit coupling. We discuss the implications of our findings, including those for the moment of inertia of neutron stars, and present the currently most precise test of general relativity’s quadrupolar description of gravitational waves, validating the prediction of general relativity at a level of 1.3 × 10<sup>-4</sup> with 95% confidence. We demonstrate the utility of the double pulsar for tests of alternative theories of gravity by focusing on two specific examples and also discuss some implications of the observations for studies of the interstellar medium and models for the formation of the double pulsar system. Finally, we provide context to other types of related experiments and prospects for the future.</p>
</blockquote>
<p><strong>Question:</strong> Is there a generally accepted set of Post-Keplerian orbital parameters with definitions?</p>
<p>To my knowledge, in a given frame there are <a href="https://en.wikipedia.org/wiki/Orbital_elements" rel="noreferrer">six timeless Keplerian orbital elements</a>, the number of which can be associated with the six elements of a state vector in a <span class="math-container">$1/r$</span> potential (see <a href="https://space.stackexchange.com/q/56024/12102">Why are there six orbital elements?</a>) plus a <em>seventh</em> parameter being a specific time or epoch.</p>
<p>The abstract says &quot;In total, we detect seven post-Keplerian parameters in this system, more than for any other known binary pulsar.&quot; Are these seven present on some pre-existing &quot;official list&quot; of post-Keplerian orbital elements, or is this paper contributing to the establishment of such a list?</p>

CC BY-SA 4.0
2021-12-14T00:31:21.507Z
A beginning of an answer is given in this article: https://link.springer.com/content/pdf/10.12942/lrr-2003-5.pdf
Pierre Paquette
CC BY-SA 4.0
2021-12-14T00:37:30.057Z
Another element of the answer is in https://journals.aps.org/prd/abstract/10.1103/PhysRevD.45.1840 (p. 45). However, both these sources list only **five** post-Keplerian parameters…
Pierre Paquette
<orbital-mechanics><general-relativity><binary-star><orbital-elements>
Post-Keplerian orbital parameters; is there a generally accepted set with definitions?
uhoh
# Post-Keplerian orbital parameters; is there a generally accepted set with definitions?

(Asked by: username_0 on 2021-12-13)

[Strong-Field Gravity Tests with the Double Pulsar](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041050) is a pretty big deal! 16 years of precision timing including "timing parallax" via VLBI has provided a dataset for which analysis requires (and *tests*) several GR effects simultaneously.

Among everything else it provides, I think this paper can be *a rich source of Stack Exchange questions and answer.*

> ABSTRACT
>
> Continued timing observations of the double pulsar PSR J0737–3039A/B, which consists of two active radio pulsars (A and B) that orbit each other with a period of 2.45 h in a mildly eccentric (e=0.088) binary system, have led to large improvements in the measurement of relativistic effects in this system. With a 16-yr data span, the results enable precision tests of theories of gravity for strongly self-gravitating bodies and also reveal new relativistic effects that have been expected but are now observed for the first time. These include effects of light propagation in strong gravitational fields which are currently not testable by any other method. In particular, we observe the effects of retardation and aberrational light bending that allow determination of the spin direction of the pulsar. **In total, we detect seven post-Keplerian parameters in this system, more than for any other known binary pulsar.** For some of these effects, the measurement precision is now so high that for the first time we have to take higher-order contributions into account. These include the contribution of the A pulsar’s effective mass loss (due to spin-down) to the observed orbital period decay, a relativistic deformation of the orbit, and the effects of the equation of state of superdense matter on the observed post-Keplerian parameters via relativistic spin-orbit coupling. We discuss the implications of our findings, including those for the moment of inertia of neutron stars, and present the currently most precise test of general relativity’s quadrupolar description of gravitational waves, validating the prediction of general relativity at a level of 1.3 × 10-4 with 95% confidence. We demonstrate the utility of the double pulsar for tests of alternative theories of gravity by focusing on two specific examples and also discuss some implications of the observations for studies of the interstellar medium and models for the formation of the double pulsar system. Finally, we provide context to other types of related experiments and prospects for the future.

**Question:** Is there a generally accepted set of Post-Keplerian orbital parameters with definitions?

To my knowledge, in a given frame there are [six timeless Keplerian orbital elements](https://en.wikipedia.org/wiki/Orbital_elements), the number of which can be associated with the six elements of a state vector in a $1/r$ potential (see [Why are there six orbital elements?](https://space.stackexchange.com/q/56024/12102)) plus a *seventh* parameter being a specific time or epoch.

The abstract says "In total, we detect seven post-Keplerian parameters in this system, more than for any other known binary pulsar." Are these seven present on some pre-existing "official list" of post-Keplerian orbital elements, or is this paper contributing to the establishment of such a list?

**Tags:** orbital-mechanics, general-relativity, binary-star, orbital-elements

> **Comments:**
> **username_1**: A beginning of an answer is given in this article: https://link.springer.com/content/pdf/10.12942/lrr-2003-5.pdf  
> **username_1**: Another element of the answer is in https://journals.aps.org/prd/abstract/10.1103/PhysRevD.45.1840 (p. 45). However, both these sources list only **five** post-Keplerian parameters…  

---

## 1 answers

### Answer by username_2 on 2021-12-14. Score: 7

I do not think there is a definitive list of post-Keplerian parameters. I would argue there is some ambiguity in the definition of a post-Keplerian parameter. Kramer et al. cite [Damour & Deruelle (1986)](https://ui.adsabs.harvard.edu/abs/1986AIHS...44..263D/abstract) ([direct PDF](https://www.ihes.fr/%7Edamour/Conferences/DamourDeruelleAIHP86.pdf)) as the source of their post-Keplerian timing model. I would call that a great source for GR effects in pulsar timing, but it doesn't use the "post-Keplerian" language.

The standard appears to be any parameter that defines a GR effect in the pulsar timing model is called post-Keplerian.

The six Keplerian orbital elements fully specify the shape of a two-body Keplerian orbit. The two-body problem does not have an exact solution in GR. Solving it requires a perturbative post-Newtonian expansion. Depending on how far one takes the expansion there are more or fewer unique parameters to measure.

In pulsar timing the model being fit is for the arrival times of radio pulses, not the orbital motion. One can infer the orbital motion from the pulse arrival times, but there's a bit more going on. For instance if a radio pulse passes nearby the pulsar's binary companion it will experience Shapiro delay, changing its time of arrival. This effect is related to the binary orbit, but doesn't strictly speaking define the orbit in the same way Keplerian orbital elements do. The Shapiro delay is certainly a non-Newtonian effect on the timing model, and the parameters that define it are grouped with the the post-Keplerian parameters.

For instance section 4 of a Living Review by [Stairs (2003)](https://link.springer.com/article/10.12942/lrr-2003-5) defines five post-Keplerian parameters:

* $\dot\omega$ - advance of periastron ($k$ is a different parameterization of the same thing)
* $\gamma\_E$ - Einstein delay amplitude
* $\dot{P}\_b$ - binary orbital period derivative
* $r$ - Shapiro delay "range" parameter
* $s$ - Shapiro delay "shape" parameter

I would say that $\dot\omega$ and $\dot{P}\_b$ clearly fit into the classification of orbital shape. The Shapiro and Einstein delay are GR effects that affect the pulse arrival times and depend on the masses and positions of the binary, but don't really define the orbital shape in the same way.

In addition to the usual five, [Kramer et al. (2021)](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041050) measure:

* $\Omega\_B^\mathrm{spin}$ - rate of relativistic spin precession of pulsar B
* $\delta\_\theta$ - relativistic deformation of the orbit

The determination of $\Omega\_B^\mathrm{spin}$ comes from the eclipsing of pulsar A by pulsar B. The orientation of B's magnetosphere affects the eclipsing, which is how it is measured. The spin precession contains a relativistic contribution, $\Omega\_B^\mathrm{spin}$, which is the non-Newtonian effect. This particular measurement is pretty unique to the double pulsar (PSR J0737–3039A/B). I don't think there is any other system where it could be measured.

$\delta\_\theta$ is related to the modification of the binary eccentricity due to GR affects. There's a second GR eccentricity modification $\delta\_r$, which cannot be measured for this system due to covariance with the pulsar spin frequency derivatives.

Damour & Deruelle (1986) state in section 3.7 that $\delta\_\theta$ and $\delta\_r$ are immeasurable due to covariances with the non-relativistic timing model, which probably explains why they are omitted from some references. They do say:

> Note however that in the long term $\delta\_\theta$ can in principle be measured when $\omega$ has changed by a sufficient amount, but previous experience with measurement of $\gamma\_E$ suggests it will be very difficult to measure $\delta\_\theta$ with any decent accuracy.

I think that Kramer et al. (2021) is the first time $\delta\_\theta$ has been measured, but I'm not 100% sure.

> **Comments:**
> **username_3**: Section 2.7.5 of [Edwards, Hobbs and Manchester 2006](https://academic.oup.com/mnras/article/372/4/1549/1186764#20550482) on the `TEMPO2` code's timing model references all 7 of the Post-Keplerian parameters. It also used the Damour and Deruelle model and references similar papers by the same people as your answer. That section also present formula for calculating the values, under the assumption that GR is correct, which is one of the tests discussed in Kramer et al 2021.  

---

