# Is there an optimum orbit for a Hipparcos/Gaia-like parallax observatory

The Gaia spacecraft is currently mapping the stars in the Milky Way by parallax. It orbits in the Earth-Sun L2 Lagrange point. The parallax observed from the spacecraft orbiting around the sun allows us to derive the distances to many stars in the milky way, as well as calculate their velocities.

Assuming no cost to placing such an observatory wherever we want around the sun, what would be the optimal orbit for optimal rate of science gains?

I apologise if this seems a big vague, let me elaborate.

If you imagine placing it in a smaller orbit, the parallax of distant stars (already very small) will be made even smaller. But information about the parallax will be returned quicker.

If you place the observatory in the orbit of Uranus, you'll have a much greater parallax and will be able to measure the distances of distant stars with much greater precision. However, it will take 80 years to complete an orbit, and the rate of distance measuring would be intolerably slow.

Intuitively, I feel like there is some obvious sweet spot that would produce measurements at an acceptable rate, though I also feel like my objective function here is ill-defined.

Would a highly eccentric orbit be preferred to a circular one? Would a particular orbit optimise the 'science gains' for one particular aspect at the expense of others (such as accuracy of nearby stars traded for accuracy of distant stars, or accuracy of stellar proper motion)? If there are multiple objectives to optimise, what would the tradeoffs be?

• Can there really be an optimum unless you know what you want to optimise? If you really need to know the parallax of a very distant object you are going to need a very long baseline and you have no choice but to wait. If you wanted to do a quick Gaia foillow-up and recheck the parallaxes of a lot of relatiuvely near stars to look for major changes, then Earth orbit is just fine. – Steve Linton Apr 15 '19 at 9:33
• One thing you could do is use multiple missions. If you sent two probes on diverging trajectories (say fast solar escape trajectories via Jupiter flypasts) you could get a very long baseline between then (at least in some directions) in a decade or so. – Steve Linton Apr 15 '19 at 9:34
• You need to set some parameters - principally, how quickly does the information need to be gathered? You also need to be clearer about what you want to measure. Just the astrometric parameters for single stars, or are you looking for photocentre motion to find binaries and planets? It's fine if you don't want to set these parameters but it might make people a bit wary of setting off writing what could be a very, very long answer indeed. And that's before we get onto the operational aspects of any planned mission. – ProfRob Apr 15 '19 at 10:56
• Yes I might refactor this question in the next couple of days. It appears different science objectives will have different optimum orbits, and so the question becomes one of multi-objective optimisation, where the relative preference for certain scientific objectives is down to individual taste. I'm not saying such a question can't be answered, but I do understand it makes it harder. – Ingolifs Apr 16 '19 at 8:34
• @Ingolifs I think it's silly for comments to put the optimization criterion on you. It might be a good idea to refactor the question and ask for the tradeoffs between a heliocentric orbit at ~1 AU (which is what a Sun-Earth Lagrange orbit really is) and an orbit at some other distance from the Sun. Asking for tradeoffs can be answered unambiguously, and answers might also provide some insight into optimization. – uhoh Oct 13 '19 at 11:40

There is several reasons why the L2 point is an optimum point for the satellite.

First you have a stable orbit with low gravitational perturbation and which is not too far away. So, you won't have a huge travel time. And the parallax is good for what the satellite is supposed to do. Secondly in a Lagrange point you can easily point out the same point. In orbit, you will have some grey area. Eventually, I think you are doing a mistake when you say : "Assuming no cost". In my opinion, if we are not sending a satellite like Gaia around another planet, it is because we want to avoid risks and sending the satellite, around Mars for example, adds a ton of risk.

That's why we use the L2 point to place a satellite in regards to all the parameters. But, you can choose another location if you want to maximize something in particular.

• This is an interesting answer but I'm not sure I understand everything. Can you explain more what the following means? "...in a Lagrange point you can easily point out the same point. In orbit, you will have some grey area." Thanks! – uhoh Apr 16 '19 at 1:46
• I think they are trying to say that out at the L2 Lagrange point, it is much easier to achieve continuous observations without being interrupted by Earth/Moon eclipses – astrosnapper Apr 16 '19 at 3:48
• @astrosnapper oh! "grey area" → eclipse, got it, thanks! – uhoh Apr 16 '19 at 4:10
• But the observations made by Gaia don't stare at one point. – ProfRob May 15 '19 at 18:26

Let's actually do the math, because it's not that hard.

Let's say you fix how many measurements you want to do per orbit (Gaia seems to do around 14), this way every possible distance $$R$$ from the sun gives you the same kind of data.

As you pointed out the precision of each individual measurements increases with $$R$$. Actually it scales directly proportional to $$R$$.

On the other hand the number $$N$$ of measurements per unit time is proportional to $$T^{-1}$$, where $$T$$ is the orbital period of the observatory. The precision of the data is proportional to $$\sqrt{N}$$, which is proportional to $$T^{-1/2}$$.

By Kepler's third law, $$T^2$$ is proportional to $$R^3$$, so the total precision is proportional to $$R T^{-1/2} = R R^{-3/4} = R^{1/4}$$ So there is no sweet spot, the precision gets better the farther you move away!

...However it does so very slowly. Uranus is about $$20\,\mathrm{AU}$$ from the sun which would result in a $$\approx 2.1$$-fold increase in precision compared to the Earth. Compared to the technical challenges, this would definitely not be worth it. This also assumes that your spacecraft is sufficiently long-lived to complete a couple of orbits and each of those takes 84 years at Uranus' distance.

To be fair, we haven't accounted for everything. A more distant orbit takes longer to complete ($$20^{3/2} \approx 84$$ times as long at Uranus), but we did the calculation assuming we still want to measure every star only 14 times. You could probably use that extra time to do more measurements, further improving precision (but probably not by a lot) or to observe many more stars.

Some other comments about Gaia: Telemetry would probably be the deal breaker here. Apparently Gaia already gathers so much data that part of the image analysis is performed on the spacecraft and only star positions are sent to earth (which of course makes it very difficult to detect errors in this part of the analysis). Transmission at any non-Lagrange point would be much worse (since the orbits would not be synchronized, the spacecraft would be on the opposite half of the solar system half of the time).

As this excellent video explains, Gaia doesn't actually do classical parallax measurements since it's not possible to know the spacecraft position with sufficient accuracy. Instead the position on sky of every star is measured repeatedly giving a overdetermined system of equations involving the position of the spacecraft which can then be solved numerically. I'm not sure if a similar method could be used for @SteveLinton's proposed mission to achieve Gaia-like precision.

• Thank you. I won't call this a complete answer (the question does ask about the rate of science gains, after all), but it does explain well some basic facts about the problem that I intuitively understood but didn't have calculations over. When I'm a bit less busy I might revisit this question. It has occurred to me that at any one time, optimum parallax data only comes from the tangent plane to the observatory's velocity vector, so out at Uranus one would be waiting a long time for parallax info from certain parts of the sky. – Ingolifs Aug 18 '20 at 22:48
• actually this is about the rate of gains since $N$ is something like a "precision rate". Of course in the long term (after a couple of orbits) .the gain in precision isn't linear but grows as $t^{1/2}$. – 0x539 Aug 18 '20 at 23:12
• Doing a fixed number of observations is an unrealistic constraint. You should consider that the measurements have to be done within a certain time. – ProfRob Aug 19 '20 at 7:20