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# Lessons Learnt: Spinning Satellites in Space | Explorer-1 Mission

### A journey to the past to understand the physics of rotating bodies through the case study of the USA's first satellite- Explorer-1.

Hello, fellow space nerds. Today, I am starting a new series: * Lessons Learnt*. This series will review past space missions. Through these case studies, we will understand some engineering lessons. For the first post in this series, we travel to times before NASA existed. A time when only one artificial object orbited our Earth: Sputnik-1. But that is not our centrepiece for today. We will keep the USA's first satellite Explorer 1 in our spotlight.

The Soviet Union launched Sputnik-1 on October 4 1957. The United States, once again, got left behind in the Space race. The government gave US Navy the go-ahead launch a satellite. Vanguard TV-3 attempted to launch US' first satellite on December 6, 1957, from Cape Canaveral. After lift-off and reaching only 1.2 m height it exploded on the launch pad. The Eisenhower administration then gave the go-ahead US army and JPL. The Jet Propulsion Laboratory designed and built the satellite within 90 days. On January 31, 1958, at 22:48 EST, the US Army Ballistic Missile Agency launched Explorer 1 using its Jupiter C rocket developed under the direction of Dr Wernher von Braun. Here is a small 5-minute documentary video on Explorer-1.

## The Problem

Any satellite in orbit needs to be oriented in a particular direction for many reasons. One simple one is to turn the antennas so that they radiate towards the Earth. Explorer-1 utilised spin-stabilisation technique to accomplish this. A rotation around the longitudinal axis was provided to the satellite. But within a few orbits, **the spacecraft changed its rotation axis**. This phenomenon was very new for scientists and engineers of that era. Let us try to understand what went wrong through simple physics.

## The Science

Many of us have studied the **law of conservation of angular momentum** for a torque-free rigid body. All of the discussion will revolve around this principle now.

### Intermediate Axis Theorem

The Intermediate Axis Theorem states that the rotation of a rigid body around the **maximum and minimum principal moment of inertia axis is stable**. This principle can be seen in action in the following video. Also known as the Tennis Racket Theorem, this medium article explains it quite well. The post explains it using Euler's equations of motion.

Coming back to Explorer-1, the satellite was rotated about the least moment of inertia axis. Therefore, being stable it should have remained rotating about the same axis.

### So, did the intermediate axis theorem fail?

The answer lies within the statement of the theorem. Explorer-1 had four flexible whip antennae sticking out of the spacecraft. Therefore, the assumption that 'the satellite is a rigid body' didn't apply. These antennae dissipated energy and led to a destabilising effect which resulted in a flat spin about the maximum moment of inertia axis.

We know that because there are no external torques, Angular Momentum is conserved.

**Momentum:**

H=Iω (constant)

While we can also write the kinetic energy of the satellite as

**Energy:**

E=½ Iω2

Emin=½ (Imaxωmin)ωmin =½ Hωmin

EMAX=½ (Iminωmax)ωmax =½ Hωmax

**Where:**

H = momentum

E = energy

ω = spacecraft angular velocity

I = momentum of inertia

With kinetic energy losing, satellite eventually went into the Emin state. Therefore, it started rotating about the maximum moment of inertia axis.

The following statement can be made:

If a spinning object has internal energy dissipation, it will orient itself so that the spin is about the axis of maximum moment of inertia (e.g., a long cylindrical object will flip into a flat spin).

Satellites with fuels will exhibit the same phenomenon as the rotating fluids will convert the kinetic energy to heat energy, much like the flexible antennae. Many experiments have been conducted onboard the International Space Station.

## Solution

We know there are many satellites which contain fuel onboard. If one of them needs to be rotated like the Explorer-1, what should be done to mitigate the problem?

The engineers at JPL solved the problem quite simply for subsequent explorer satellites. Two **fibreglass slot** antennae replaced the original flexible antennae. Now, the Explorer-3 spacecraft indeed was a rigid body. Therefore, thanks to Intermediate Axis Theorem, the rotation about the minimum moment of inertia axis was stable.

But for satellites with fuel, we are still to solve the problem. The solution for this is also simple enough. The difficulty arises due to the dissipation of kinetic energy. If we could somehow **impart energy to the spacecraft**, the problem would again be solved. Hence the engineering solution, Reaction Wheels came into existence. Reaction wheels are a large moment of inertia bodies which can be rotated by converting electrical energy (motors).

## Summary

In summary, a rigid body rotating about the highest or least moment of inertia axis is stable. If it starts losing Kinetic Energy through internal dissipation (Angular Momentum is conserved), the body will tend to rotate about the maximum moment of inertia axis only.

The following video by Veritasium explains this whole phenomenon quite well.