# Ion Propulsion Math

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## Introduction

Figure 1: Dawn Mission Profile. (Source)

NASA has a project known as Dawn that put a space probe in orbit around the asteroids Vesta and then CeresCSPAN presented an excellent Dawn mission briefing given by Marc Rayman, the Mission Director and Chief Engineer.  One of the most interesting aspects of the  Dawn spacecraft is its use of an ion thruster to maneuver it from one destination to another. This post presents some simple math that can be used to determine some of its key performance characteristics.

This briefing was for the general public and presented some excellent material. For those who like to look at my raw files, I include my Mathcad, PDF, and XPS versions here.

## Background

### Dawn Spacecraft Information

The following quote from the Wikipedia article on the Dawn spacecraft provides lots of data for me to use in my analysis.

The Dawn spacecraft is propelled by three xenon ion thrusters … and uses only one at a time. They have a specific impulse of 3,100 s and produce a thrust of 90 mN. The whole spacecraft, including the ion propulsion thrusters, is powered by a 10 kW (at 1 AU) triple-junction gallium arsenide photovoltaic solar array manufactured by Dutch Space. Dawn was allocated 275 kg (606 lb) of xenon for its Vesta approach, and carried another 110 kg (243 lb) to reach Ceres, out of a total capacity of 425 kg (937 lb) of on-board propellant. With the propellant it carries, Dawn can perform a velocity change of more than 10 km/s over the course of its mission, far more than any previous spacecraft achieved with onboard propellant after separation from its launch rocket.

I can summarize this data with the following list:

• The thruster provides a thrust of 90 mN
• The thruster generates a specific impulse of 3,100 s.
• The thruster and fuel supply can provide a total velocity change ΔV= 10 km/s.
• The photovoltaic array has a maximum power output of 10 kW @ 1 AU.
• fuel mass of 425 kg.

I also found some additional information useful:

•  xenon molecules at a velocity of 30 km/s. (Source)
• maximum ion thruster input power level of 2.3 kW. (Source)
• ion thruster power efficiency of 61%. (Source)
• Dawn spacecraft Beginning of Life (BOL) mass of 1240 kg. (Source)

### Dawn Thruster Block Diagram

Figure 2 shows a block diagram of Dawn's ion thruster. The basic construction is  reminiscent of the electron gun in an old tube-type television.

The thruster concept is simple:

• supply xenon from the propellant supply bottle to the ionization chamber
• ionize the xenon
• accelerate the xenon ions using a high-voltage electric field

Figure 2: Dawn Thruster Block Diagram. (Source)

## Analysis

### Objective

My plan is to compute the following spacecraft characteristics:

• thrust (FIon)
• fuel burn rate (m')
• mission time (TMission)
• ΔV

based on the

• array power (P)
• xenon molecular velocity (v)
• fuel quantity (MFuel)
• spacecraft mass (MBOL)
• engine efficiency (η).

### Solution Setup

Figure 3 shows how I setup this calculation.

Figure 3: Analysis Setup.

### Determine Key Performance Indices

Figure 4 shows how I used the conservation of energy to derive formulas for the thrust and fuel burn rate. The analysis approach is simple:

• Solar power (minus efficiency losses) is converted from photon energy to xenon ion energy.
• The kinetic energy of the xenon ions is converted to a velocity.
• The velocity and mass loss rate is then converted to a change in momentum.
• The change in momentum is equivalent to force.

All the numbers obtained using this approach reasonably match the values quoted in the Wikipedia. I should note that the rated mission lifetime is 50K hours, but there is only enough fuel for 40K hours at full power (i.e. 1 AU from the Sun). The 40K hours is actually reasonable because the Dawn spacecraft will spend a significant amount of time in orbit about these Vesta and Ceres, so it will not have the thruster on all the time.

Figure 4 Mathcad Note: You will see that many of the variables in Figure 4's derivation are teal-colored rather than black-colored. For derivations, I often use a different variable style to prevent an early numeric definition from interfering with a downstream symbolic derivation. A different variable style in Mathcad means that a variable with the same name as a another style is treated as a different variable. Otherwise, Mathcad will substitute the upstream numeric value of the variable, which I do not want when I need a symbolic result.

Figure 4: Determination of Four Key Engine Parameters.

## Conclusion

I was able to use some simple math and physics to reproduce some of the key performance metrics for the Dawn spacecraft. This helps me understand how ion propulsion works and where it would be useful to use. Because it requires so much power, I can see where it would primarily be useful for spacecraft that are close enough to the Sun to use solar power. A radioactive battery (often used on deep space probes) would not provide sufficient power to make an ion drive work – Rayman makes this comment in his briefing.

The Dawn mission was very interesting. I want to commend Marc Rayman for the excellent briefings he has presented throughout the project. I have enjoyed keeping up with their discoveries about Vesta and Ceres.

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### 3 Responses to Ion Propulsion Math

1. Filip De Somer says:

Dear Mark, I am curious how you are able to show the symbolic result after the eq1, eq2, eq3 vector. When I do not see any equation vector after the arrow although they seem to be assigned. What is the trick?

• mathscinotes says:

Hi Filip,

First, I have now added my source to this post, which shows my trick. Sorry that I often forget to include my source – I write these posts over lunch at work, and I am so focused on writing that I often forget to include the source.

Mathcad 15 has a number of problems that are irritating – this is one of them. It often will not show the symbolic results, even though they are there. If you look at my source, you will see that I include the following statement off to the side of my derivation.

I generally do not include this statement in my screen captures. This statement should not be necessary but I often include it to ensure that I can see the results of a derivation. When I include this statement after a derivation, I usually see my results.

Sometimes, I still do not see my result. Then go to the Find statement and just move it up and down. This action nearly always shows the result. This is screwy, but I am so used to it that I do not think about it anymore.

mark

2. Filip De Somer says:

Thanks, this is a nice trick that I didn't knew. Especially, moving the Find is not intuitive. Filip