M060 · Orbital attitude / internal momentum exchange

Satellite Attitude & Gravity Gradient

Build a rigid spacecraft from a uniform core and two equal boom-tip masses. Vary its inertia, orbital altitude and initial tumble. Compare free rotation with gravity-gradient libration, then command a finite reaction wheel and measure the saturation budget.

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 $\text{Pending}$

PREPARE / COMPARE / VERIFY

Prepare the rigid body

Show angular momentum and axesArrows have independently normalized lengths. Dimensions share one scale; Earth and orbit are inset 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. Initial angular velocity follows the orbital frame before the two body-axis kicks are added. The wheel rotates about the third body axis; its constant acceleration command lasts at most one quarter orbit and stops earlier at capacity. This is an open-loop command, not a pointing controller. The local rate estimate applies only to planar, unforced infinitesimal motion about alignment.

Prescribed circular Earth orbit with a central spherical gravity field and the leading gravity-gradient torque. Body first axis points outward at alignment; orbit normal is world third axis. Core plus symmetric point masses sets the inertia; rods, face markings and the displayed wheel are massless schematic elements. The ideal wheel carries relative momentum along the third body axis with negligible added locked inertia; wheel kinetic energy, efficiency and power limits are not modeled. Reported kinetic energy and motor work concern the body, not the complete wheel battery system. The rotating-frame energy ledger accounts for prescribed orbital rotation and motor work. No drag, magnetic field, damping, flexible boom, eccentricity, orbit feedback, closed-loop control or momentum unloading. A gravity-gradient boom produces libration without dissipative settling; an internal wheel cannot change total world momentum when external torque is off. Orbit count sets dimensionless time; altitude changes physical period and SI rates. Uniform scene magnification is automatic; vector lengths are normalized.

Start with the essentials

Focus question: Which inertia arrangement supports radial pointing, and what happens when a wheel saturates?
One-sentence intuition: Gravity gradient applies an external torque; a wheel redistributes internal momentum. Neither mechanism supplies dissipative settling in this model.

Core mathematical model

Orbit time scale

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

A higher orbit slows the same dimensionless attitude trajectory in physical time. This model prescribes the circular orbit.

Boom inertia

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

The core has mass fraction one minus the tip fraction; half the tip mass sits at each symmetric endpoint.

Gravity-gradient torque

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

Outward radial and inward nadir give the same torque because both factors change sign. The approximation requires body size much smaller than orbit radius.

Ideal wheel momentum exchange

\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 changes internal momentum distribution. With gravity torque off, total world momentum is fixed even during a command.

Planar alignment limit

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

For zero roll and no wheel command, radial alignment librates if the second inertia exceeds the first. This scalar limit does not establish full three-dimensional stability.

Rotating-frame work ledger

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

Subtract integrated motor work on the body to obtain the conserved ledger. Potential has an arbitrary attitude-independent offset. Wheel energy and battery work are omitted.

Common difficulties

A boom eventually stops the motion

Typical misconceptionA stable gravity-gradient orientation must settle automatically.

Better mental modelWithout damping, conservative libration persists. A planar restoring rate alone does not certify full 3D stability.

A wheel deletes momentum

Typical misconceptionSpinning up an internal wheel removes total satellite momentum.

Better mental modelThe body and wheel exchange momentum; an external torque is required for unloading.

Run the experiment

01
Inspect the boom
Compare libration and no-boom presets; inspect overview, side and top views.
What to observe: Point masses change two principal inertias and the gravity-gradient response.
02
Separate orbital and physical time
Compare the default and higher-orbit presets at the same playback fraction.
What to observe: Dimensionless attitude agrees, while physical elapsed time and angular rates differ.
03
Command the wheel
Disable external torque, apply a wheel command and compare total momentum with wheel momentum.
What to observe: The body responds, but total world momentum stays fixed; the energy change follows motor work.
04
Reach capacity
Select saturation and review the full record, then add a three-dimensional tumble.
What to observe: The command stops at capacity. Removing stored momentum needs external torque; no unloading mechanism is modeled.