F-X4 · A clock in a changing orbit

Pulsar Timing and Orbital Decay

How can pulse arrival times reveal an orbit that slowly shrinks? A shortening orbital period brings each periastron slightly earlier than a fixed-period prediction. These small advances add up into a curved trend over decades, which precise pulse timing can measure.

Tilt a binary pulsar orbit and follow the changing light-travel time. Then compare decades of synthetic periastron times with the prediction for a constant orbital period.

Projected orbital size $1$

Accumulated time shift $1$

Fitted period derivative $1$

GR period derivative $1$

All plotted observations are synthetic. The orbit uses a Kepler ellipse with the period and component masses reported for PSR B1913+16 in Weisberg and Huang (2016); inclination and eccentricity are adjustable teaching parameters. The orange curve is the geometric Roemer light-travel delay. The long-term panel fits a quadratic periastron-time shift with a constant period derivative, including offset and linear terms. Its deterministic noise is illustrative, not an observing-error model. The displayed GR prediction is the leading quadrupole orbital-period derivative for the selected eccentricity. The two trends are separate views of timing, not a full pulse-arrival fit. Galactic acceleration, proper motion, dispersion, spin evolution, Einstein and Shapiro delays, periastron advance and beam precession are omitted. Real gravity tests must correct the observed period derivative for kinematics. Stellar size, beam width and animation speed are schematic; orbital shrinkage is not magnified into the animation.

Synthetic timing observations

One orbit and decades of timing

Light-travel delay over one orbit

\text{orbital phase},\quad\Delta_R/\mathrm{s}

Orange: geometric light-travel delay, with the constant distance to the system removed. Phase zero is periastron.

Periastron against a fixed-period prediction

t/\mathrm{yr},\quad\Delta T/\mathrm{s}

Pink: synthetic times with illustrative noise. Teal: the injected quadratic trend. Gray: a constant-period prediction. A negative shift means an earlier periastron.

Read the discoveries behind these models →

Start with the essentials

Focus question: How can pulse arrival times reveal an orbit that slowly shrinks?
One-sentence intuition: A small decrease in every orbital period accumulates into a time shift that grows with the square of the observing span.

Core mathematical model

The geometric light-travel delay

\Delta_R=z/c

The coordinate z points away from the observer. A positive displacement adds light-travel time. The plot subtracts the constant distance to the binary.

Accumulated periastron-time shift

\Delta T\simeq\frac{1}{2}\frac{\dot P_b}{P_b}t^2

The fixed-period ephemeris predicts periastron every P_b seconds. A small constant derivative changes that period gradually; negative derivative makes periastron earlier. This first-order approximation requires the period change to be small over the observing span.

Common difficulties

Two kinds of early arrival

Typical misconceptionEvery early pulse means that the orbital period has shortened.

Better mental modelMotion around a fixed orbit also changes the light path. The periodic delay and the decades-long curvature have different shapes and must be fitted separately.

Run the experiment

01
Tilt the orbit
Choose Nearly face-on, then drag across the scene to increase Orbital inclination.
What to observe: The light-travel delay grows as more of the orbit points along the line of sight. The long-term periastron drift stays the same.
02
Wait twice as long
Choose Shrinking orbit, then compare observing spans of twenty and forty years.
What to observe: Twice the span gives four times the accumulated advance. The orbit itself barely changes on this timescale.
03
Remove the period change
Choose Fixed period and compare the synthetic points with the zero line.
What to observe: The long-term curvature disappears. The periodic light-travel delay remains because the pulsar still moves around its orbit.