# Ion propulsion mathematics

-January 31, 2018

During a visit to NASA’s Jet Propulsion Laboratory (JPL), I had the privilege of seeing a new experimental ion propulsion rocket engine test.

A Xenon ion propulsion engine (Image courtesy of NASA)

NASA has been using ion propulsion engines for many, many years and commercial satellites use them too! Such rocket engines are not science fiction.

But, here on my Math Is blog, I want to discuss the mathematics that helped create these engine designs and are being used to develop more advanced ones for the future.

How do charged particles gain their speed in an electric field?

Let’s look at some physics math for the kinetic energy of a particle:

Kinetic energy (KE) = ½ mv2 we know this from our intro physics classes where m is mass and v is velocity.

E = qV gives us the energy of a charged particle where q is the charge and V is the potential difference in Volts in which the particle falls and gains energy.

Let’s take an ion propulsion engine example:

The NSTAR ion engine was developed for the Deep Space 1 satellite. This engine uses Xenon atoms that have a mass m = 2.2 x 10-25 kg and a charge of the all-familiar q of 1.6 x 10-19 coulombs, a constant that we all know and have indelibly printed into our engineering brains.

We know that 1/2mv2(Kinetic energy) = q (charge) x V (Voltage)

So, now let’s calculate the speed of a Xenon atom in km/hr, assuming that the voltage grid V of the ion engine is 1,300 V:

Let KE = E and we now solve for v:

Kinetic energy = charge x Voltage

v2 = 2qV/m = 2(1.6 x 10-19coulombs)(1,300V)/(2.2 x 10-25kg) = 2.0 x 109

Taking the square root of 2.0 x 109 gives us v = 44.2 km/sec or 159,000 km/hr WOW! That’s fast.

Steve Taranovich is a senior technical editor at EDN with 45 years of experience in the electronics industry.

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