M060 · Satellite pointing and momentum exchange

Satellite Attitude & Gravity Gradient

What can a long boom and a momentum wheel each do to keep a satellite pointed? A boom can keep the satellite pointing roughly at Earth: gravity pulls slightly harder on the near end, which gives a restoring torque when the mass is arranged right. With nothing to damp it, the satellite keeps rocking about that direction. A wheel can only move momentum between itself and the body, until it hits its limit; it cannot remove any. Shedding total momentum takes an outside torque.

Build a rigid spacecraft from a uniform core and two equal masses at the ends of a boom, then change its inertia, orbital altitude and initial tumble. Compare free rotation with the swinging caused by the gravity gradient, then drive a reaction wheel and measure how much momentum it absorbs before it saturates.

Physical elapsed time $\text{Pending}$

First principal inertia $\text{Pending}$

Second principal inertia $\text{Pending}$

Third principal inertia $\text{Pending}$

First body angular velocity $\text{Pending}$

Second body angular velocity $\text{Pending}$

Third body angular velocity $\text{Pending}$

Selected-axis deviation $\text{Pending}$

Body rotational energy $\text{Pending}$

Total angular momentum $\text{Pending}$

Prepared state $\text{Pending}$

Local growth / libration rate $\text{Pending}$

Max scaled balance defect $\text{Pending}$

Two-step attitude difference $\text{Pending}$

Circular orbit period $\text{Pending}$

Attitude-dependent potential $\text{Pending}$

Rotating-frame energy $\text{Pending}$

Motor work on body $\text{Pending}$

Signed wheel momentum $\text{Pending}$

Wheel momentum capacity $\text{Pending}$

Wheel command state $\text{Pending}$

Set up, compare, then verify

Prepare the rigid body

Show angular momentum and axesEach kind of arrow is normalized on its own. The body dimensions share one scale; Earth and the orbit appear only as an inset for context.Total spacecraft mass

m

100.00\,\mathrm{kg}

Core first-axis dimension

a

0.80\,\mathrm{m}

Core second-axis dimension

b

1.20\,\mathrm{m}

Core third-axis dimension

c

0.60\,\mathrm{m}

Centre to each tip mass

\ell

4.00\,\mathrm{m}

Total tip-mass fraction

f

0.20\,

Circular orbit altitude

H

500.00\,\mathrm{km}

Initial in-plane pointing angle

\theta_0

25.00\,{}^\circ

Initial roll about first axis

\rho_0

8.00\,{}^\circ

Extra third-axis rate

\delta\omega_3/n

0.00\,

Extra first-axis rate

\delta\omega_1/n

0.00\,

Gravity-gradient torque switch

s

1.00\,

Wheel acceleration command

u/(I_3n^2)

0.00\,

Wheel momentum capacity

h_{\max}/(I_3n)

1.00\,

Record duration in orbits

nT/(2\pi)

2.00\,

Reviewed fraction

t/T

0.00\,\%

Drag the orange pointing handle. The initial angular velocity first matches the rotation of the orbital frame; then the two kicks about body axes are added. The wheel spins about the third body axis. Its constant-acceleration command lasts at most a quarter orbit and stops earlier if the wheel reaches its limit. The command is open-loop; there is no pointing controller. The local rate estimate holds only for infinitesimal, unforced motion in one plane about the aligned state.

The satellite follows a prescribed circular orbit in the central gravity field of a spherical Earth, and the gravity-gradient torque is kept to leading order. When aligned, the first body axis points radially outward; the orbit normal is the third world axis. The core and the two symmetric point masses set the inertia; the rods, face markings and drawn wheel are massless sketches. The ideal wheel stores momentum relative to the body along the third body axis and adds negligible locked inertia; its kinetic energy, efficiency and power limits are not modeled. The kinetic energy and motor work shown refer to the body, not to the whole wheel and battery system. The energy balance in the rotating frame includes the prescribed orbital rotation and the motor work. Drag, magnetic fields, damping, boom flexing, orbital eccentricity, orbit feedback, closed-loop control and momentum unloading are left out. So the boom makes the satellite swing without settling, and with the external torque off the internal wheel cannot change the total angular momentum in the world frame. The number of orbits sets the dimensionless time; the altitude changes the period in seconds and the rates in SI units. The scene is magnified uniformly and automatically, and arrow lengths are normalized.

Start with the essentials

Focus question: Which mass layout lets a satellite keep pointing along the radial direction, and what happens when its wheel saturates?
One-sentence intuition: The gravity gradient is an external torque, while a wheel only moves angular momentum between itself and the body. In this model neither one dissipates energy, so nothing settles on its own.

Core mathematical model

Orbital rate and period

n=\sqrt{\frac{\mu}{(R_E+H)^3}},\qquad P=\frac{2\pi}{n}

In a higher orbit, the same attitude motion counted in orbits takes longer in seconds. The circular orbit is prescribed; the model does not compute it.

Inertia with boom tip masses

I_1=I_{1,c},\quad I_2=I_{2,c}+fm\ell^2,\quad I_3=I_{3,c}+fm\ell^2

A fraction of the total mass sits at the boom tips, half at each end, and the core carries the rest.

Gravity-gradient torque

\boldsymbol\tau_b=3n^2\hat{\mathbf r}_b\times I\hat{\mathbf r}_b

Using the outward radial direction or the inward direction toward Earth gives the same torque, because the unit vector appears twice and both signs flip. The formula holds when the satellite is much smaller than the orbit radius.

Momentum exchange with an ideal wheel

\mathbf L=R(I\boldsymbol\omega_b+h\mathbf e_3),\quad\dot{\mathbf L}=R\boldsymbol\tau_b,\quad\dot h=u

The wheel only moves angular momentum between itself and the body. With the gravity torque off, the total angular momentum in the world frame stays fixed, even while the wheel is being driven.

Planar swing about the radial direction

\ddot\theta+\frac{3n^2(I_2-I_1)}{2I_3}\sin(2\theta)=0

With no roll and no wheel command, the satellite swings back and forth about the radial direction when the second moment of inertia is larger than the first. This one-angle equation does not prove that the full three-dimensional motion is stable.

Energy balance in the rotating frame

J=\frac12\boldsymbol\omega_b\cdot I\boldsymbol\omega_b+\frac32 n^2\hat{\mathbf r}_b\cdot I\hat{\mathbf r}_b-nL_z,\quad\dot J=-u\omega_3

This quantity changes only through the motor's work on the body; subtract that accumulated work and the result stays constant. The potential energy may be shifted by any constant that does not depend on attitude. The wheel's own energy and the battery's work are left out.

Common difficulties

The boom will eventually stop the swinging

Typical misconceptionIf the gravity-gradient orientation is stable, the satellite should settle into it by itself.

Better mental modelWith no damping, energy is conserved and the swinging keeps going. A restoring torque in one plane is also not enough to prove the full 3D motion is stable.

A wheel can get rid of momentum

Typical misconceptionSpinning up the internal wheel takes away the satellite's total angular momentum.

Better mental modelThe body and the wheel only trade angular momentum. Getting rid of it requires an external torque.

Run the experiment

01
Look at the boom
Compare the Boom libration and Remove the boom presets, and look at the overview, side and top views.
What to observe: The tip masses change two of the principal moments of inertia, and the gravity-gradient motion changes with them.
02
Separate orbits from seconds
Compare the default and Higher circular orbit presets at the same playback position.
What to observe: Counted in orbits, the attitude is the same. In seconds, the elapsed time and the angular rates are different.
03
Drive the wheel
Turn off the external torque, apply a wheel command, and compare the total angular momentum with the wheel's.
What to observe: The body's rotation changes, but the total angular momentum stays fixed. The energy change matches the work done by the motor.
04
Saturate the wheel
Choose the Wheel saturation preset and replay the whole run, then add a three-dimensional tumble.
What to observe: The command stops once the wheel reaches its momentum limit. Only an external torque could remove the stored momentum, and the model has no way to unload it.