---
title: "Tidal Locking & Libration Interactive Lab | Sandbox Physics"
description: "The same face. Still turning. Why does always showing one face require continued rotation?"
canonical_url: "https://sandboxphysics.com/labs/tidal-locking/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/tidal-locking/"
last_modified: "2026-10-18"
---

M080 · Spin meets orbit

# Tidal Locking & Libration

Spin an elongated satellite, add dissipation and follow its marked face around an orbit. Compare circulation, libration and synchronous rotation against an explicit energy account.

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

Planar rigid-body gravity-gradient approximation on a prescribed circular orbit. Phenomenological damping, normalized time and enlarged satellite; no orbit migration, material rheology or lunar capture timescale.

Physics tutorial

## Losing relative spin

**Background** Gravity acts differently across an elongated body, making a torque that favors radial alignment.

**Why it matters** Connect a spatial picture to a quantitative model you can test.

Start with the essentials

- **Focus question**: Why does facing the planet require continued rotation?
- **One-sentence intuition**: Synchronous rotation means one inertial turn per orbit. Zero relative spin is not zero inertial spin.

### Core mathematical model

#### Time and orientation

$$
u=nt,\quad\delta=\theta-u,\quad\frac{\omega}{n}=1+\delta'
$$

A prime is a derivative with respect to normalized orbital time.

#### Torques and motion

$$
\delta''=-\frac32\eta\sin(2\delta)-\gamma\eta\delta',\qquad\eta=\frac{B-A}{C}
$$

The gravity-gradient term follows a rigid-body quadrupole expansion. The damping term is phenomenological.

#### Relative energy and dissipation

$$
\mathcal E=\frac12(\delta')^2+\frac34\eta[1-\cos(2\delta)],\quad Q'=\gamma\eta(\delta')^2,\quad(\mathcal E+Q)'=0
$$

All terms are normalized by spin inertia times orbital angular rate squared. The chosen potential is zero at radial alignment.

#### Shape and small libration

$$
\frac ba=\sqrt{\frac{1-\eta}{1+\eta}},\qquad T_u=\frac{2\pi}{\sqrt{3\eta}}\quad(\gamma=0,\ |\delta|\ll1)
$$

The displayed homogeneous ellipsoid has this axis ratio. Its small-amplitude period gives an independent numerical check.

### Common difficulties

#### A torque does not guarantee capture

**Typical misconception** Gravity automatically stops relative rotation.

**Better mental model** Switch off dissipation: relative mechanical energy is conserved and the body can keep circulating or librating.

### Run the experiment

1. 01

   #### Follow one face

   Choose Already synchronous and play one orbit. Watch the gold spot in Orbit and Radial views.

   **What to observe:** The body turns in inertial space while staying fixed relative to the radial line.

2. 02

   #### Compare equal starts

   Run With dissipation, then No dissipation. Scrub to the end and compare the energy chart.

   **What to observe:** Dissipation transfers relative mechanical energy into the accumulated loss account. Without it the oscillation or circulation persists.

3. 03

   #### Check the model

   Try Spherical body, then halve the integration step in a nonspherical case.

   **What to observe:** A sphere has neither modeled torque nor damping. A smaller step reduces the numerical energy-budget residual.

Physical conclusion

Gravity alone can sustain libration or circulation. The specified dissipation removes relative mechanical energy.

Primary sources: [MIT Press · Spin–orbit coupling](https://mitp-content-server.mit.edu/books/content/sectbyfn/books_pres_0/9579/sicm_edition_2.zip/chapter002.html) [NASA · Tidal locking](https://science.nasa.gov/moon/tidal-locking/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Measure gravity, split motion into components, add contact forces and collisions, then carry the same laws into 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)