---
title: "Pulsar Timing and Orbital Decay Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/pulsar-timing-residuals/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/pulsar-timing-residuals/"
last_modified: "2026-10-24"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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 →](https://sandboxphysics.com/stories/astronomy-gravity/)

Physics tutorial

## Two patterns in pulsar arrival times

**Background** A pulsar sweeps a radio beam across Earth as it spins. In a binary, the source moves toward and away from us, changing how far each pulse travels.

**Why it matters** An orbital period changes by only a tiny amount each year. Adding many orbits turns that small change into a measurable early arrival.

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 misconception** Every early pulse means that the orbital period has shortened.

**Better mental model** Motion 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

1. 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.

2. 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.

3. 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.

Physical conclusion

The shape and timescale of a timing trend tell us which physical effect to test.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)