# CubeSat Orbit Determination

I'm thinking about the feasibility of making a CubeSat to test the EM drive thruster. Assuming the microwave electronics and resonant cavity could be scaled down to CubeSat size (5 watts, let's say), and we expect the most recently NASA reported ~1 millinewton per kilowatt, my quick calculations (check my math if they seem off) say that we'd expect to see a delta-V of around 3 m/s over a month's time, and thus a resulting orbit altitude increase/decrease of 5km.

Of course, there's also drag acting on the spacecraft, so you'd probably have to fire it for month in one direction, and then a month in the other, to identify if there's delta-V independent to that of drag.

So, for this experiment to be useful, we'd need a way to pretty accurately measure the orbit of the satellite over time. How is this done normally? I'm imagining things like ground-based radar, or perhaps just tracking the communications radio, or maybe some sort of onboard computer and sensors, but I'm not sure if any/all of those are feasible or accurate. Is there a way to get enough accuracy that this experiment seems feasible?

• "Assuming the microwave electronics and resonant cavity could be scaled down to CubeSat size (5 watts, let's say)" It is not just the wattage that might have a lower limit. The EM drive uses a Microwave cavity which itself depends upon the wavelength of the microwave EM radiation used. – Andrew Thompson Dec 28 '16 at 16:22
• – Puffin Dec 28 '16 at 19:19
• Just in case you are completely new to the last links - the US gov, through NORAD, routinely measures the positions of anything big enough to track, which includes cubesats if they are in LEO, and allows some websites to publish a subset in a "good enough for many purposes" format called a TLE. Try celestrak.com or heavens-above.com. Each satellite is assigned both a NORAD number and an international designator shortly after launch. – Puffin Dec 28 '16 at 19:25
• Awesome, thanks! I wasn't familiar with the NORAD tracking, so thanks for pointing me in the right direction. – zplizzi Dec 28 '16 at 23:21

## 2 Answers

I'll try to answer your immediate question first, then propose some alternate and more sensitive variations.

Let's start with a cubesat (maybe 3U) with a mass of 5kg and a miniaturized EM drive thruster (first peer-reviewed EM drive paper and pdf thereof) has been put inside, or maybe once in orbit the 3U "pops open" and a folded resonator made of a springy material with conductive inner coating expands to a nicer resonator size. One way or the other, let's assume the TM212 mode resonates properly with a high Q approaching something like the 7000 indicated in Figure 4 of that paper. That's pretty high, but let's just go with the flow. It does (potentially) violate basic physical laws anyway.

Ignoring the issue raised in @AndrewThompson 's comment about the thrust potentially scaling with the size of the cavity, let's use a simply scaled propulsive force for 5 Watts:

0.005 kW $\times$ 1 mN/kW = 5E-06 Newtons.

Since the force is so tiny, we should at least try to avoid as much atmospheric drag as possible. Let's choose an initial circular orbit with altitude of 800 km. Using an equatorial radius of 6378 km. the initial semi-major axis will be (in meters):

$$a_i = (6378 + 800) \ \times \ 1000$$

Using the vis-viva equation, setting r = a for a circular orbit and using a value of 3.986004418E+14 m^3/s^2 for the standard gravitational parameter of Earth $GM_e$, the initial orbital speed will be:

$$v_i = \sqrt{GM_e \ / \ a}$$

or about 7451.9 m/s.

Starting with some basic Newtonian physics:

$$\Delta p = Force \times time$$

$$\Delta v = Force \times time \ / \ mass$$

Five micro-Newtons for 30 days against a mass of 5kg gives a $\Delta v$ of 2.6 m/s, so yes your quick calculation is good.

Now a really interesting result. This illuminating answer by @MarkAdler confirms that, for a good rule-of-thumb for a very low tangential-only accelerations resulting in a gradual spiral between two co-planar circular orbits, the change in orbital speed will be negative the total $\Delta v$. So in the limit of zero thrust and infinite time, a total $\Delta v$ in the tangential direction of motion will actually slow the space craft's orbital speed by $\Delta v$ while simultaneously raising the altitude of the orbit!

So that gives a final orbital velocity:

$$v_f = v_i - \Delta v$$

or 7451.9 - 2.6 = 7449.3 m/s. Using the vis-viva equation again "in reverse" the final semi-major axis will be:

$$a_f = (GM_e / v_f)^2$$

or about 7183.0 km, an increase of 5 km. The period of a circular orbit is just the circumphrence divided by the speed:

$$T = 2 \pi a / v$$,

so the period will change by 6 seconds, from about 6052 to 6058 seconds, which means the positions of cube sats in the two orbits would drift an additional 42 kilometers apart after every single 100 minute orbit!!

So actually, de-phasing will be a far more sensitive method than looking for an altitude change (as I've already mentioned here), and without considering all the other problems, it should be sensitive to $\Delta v$'s much lower than this.

So you can really turn the EM drive ON for one day, then off for one day, then repeat, and as long as you are tracking it by radar several times a day, some kind of test could be performed.

I am going to briefly suggest modifications to consider.

• deploy a pair of matched cubesats, have them measure their relative positions by time-of-flight (light or radio) and have them alternate one OFF, the other ON every few hours or few days, depending on details.

• give them GPS capability, but use a commercially available, orbit-qualified cubesat GPS unit, and not somebody's phone!

• connect a pair together with a tether, or tether one to a dummy ballast, and (try to) get them to spin around their center of mass, or at least to deviate from their equilibrium position in orbit (the basis of an interesting follow-up question??)

As some of the comments have mentioned space-track provides reasonably upto date TLEs of anything they can track (including cubesats), although I wouldn't think this would be the best option for your use.

GPS trackers are now pretty accurate, from experience the error rates in TLEs don't come close to the accuracy you would get from a GPS tracker on your cubesat. The problem with using GPS on a satellite is that most GPS chips are specifically built to make it harder for rogue nations to make course correcting missiles.... no, seriously.

The GPS is your phone and in your car sat nav won't work if you either travel faster than 1,900 km/h or higher than 18 km altitude. LEO satellites travel at around 28,000 km/h and are at least 300 km in altitude. These are the CoCom limits. The rational behind this is that you can't retro fit a missile with an iPhone and then use the GPS system to get the missile position for autonomous course correction (amoung other, less cool things I guess).

The good news is that it's not all the hard to get a GPS chips without these CoCom limits, I'm pretty sure you end up on a list somewhere (NSA, etc.) but hey, if you're on the space exploration stack exchange then you've probably got at least 1 very dangerous thing just sitting in your desk (rocket fuel, mini rail gun etc.) so you're already on a few lists. Actually I'm not sure this was good news after all.

Edit: The use of continous thrust would change an orbit. Whether or not this would be negligible is another question. You're talking about 0.001 N/Kw and a power of 5 Watts, so a total thrust of 0.000005 N. Now over time this would make a change in velocity significant enough to get you to leave Earth orbit if there were no other pertubations. A back of the envelope drag calculation gives 0.000075 N/m2; for a cubesat that is 10x10 cm you'd have a drag force of 1/6th of the force your trying to measure. The bad news is that the drag force calculations are very rough and hugely altitude dependant, you could easily double/half the number by changing your altitude from my test case of 400 km. The good news is the higher your altitude the less of an effect drag will have. The key thing here is that there are many pertubations and your small force could convieveably get lost in the noise. Two cubesats, one with and one without the drag device would likely provide more conclusive evidence!

As a side note - be careful when using TLEs for propagation. You must use the simplified perturbation models to propagate TLE data forward or your results will be incorrect!

• Interesting answer. Can you add a few helpful links to back up some of your points? For example, the numerical values of the limits on speed and latitude look wrong, and so does your spelling of the limits. If someone wanted to read more, there should be a link showing where you've gotten these. Also, while TLEs have limits, over time a continuous thrust will change an orbit enough to be noticeable. For example a change in altitude will show as a change in phasing as well, all you have to do is wait. Can you support your "it's not at the hard to...", and mention you can buy cubesat GPS units? – uhoh Jan 20 '17 at 2:39
• It is also good practice to include links to related questions and answers here if they contribute to your answers. Why don't you look through the (currently) 24 questions in this stackexchange that are tagged gps and see if you find something to link to that is helpful to future readers. You might also just look at the comments under the question to see links to other question and answers that address TLE accuracy. – uhoh Jan 20 '17 at 2:46
• I'd love to add some helpful links. Unfortunately the SE interface isn't too great on the mobile app and it often fails to add the actual links! Any suggestions? – user6916458 Jan 21 '17 at 2:20
• I would say rounding by a 2% is probably OK when the purpose of the statement is to make it clear that the satellites in LEO travel an order of magnitude faster. It's been discussed before on this site and fro. What I remember the concensus is to not suggest and edit to an answer when it doesn't add specifics to the question. I have also provided a link to the CoCom limits if someone was all that interested. Either way, 1852 or 1900 km/h are a far cry from LEO orbit speeds which was the point I was making! (BTW one of your previous comments still isn't showing up). – user6916458 Jan 21 '17 at 12:15
• A 2% (or even 10%) error doesn't bother me - we're throwing around numbers with much less precision already in this question. What would have been really helpful, though, would have been some numbers comparing the accuracy of GPS and TLE's. – zplizzi Jan 22 '17 at 19:34