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

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

It may be the Jupiter-Sun L2 point. There is not light from Sun. This provides a minimum low temperature. This is necessary for a minimum noise and for a greater sensitivity. There is acceptable orbit period. Which provides 3...5 passages between an opposite points of the orbit in 2...4 decades including flight from Earth to Jupiter. This is close to the time of work of some large space astronomical observation missions. Several passages are necessary to an accurate definitions of distinctions of parallaxes from own movements of stars.

But this requires solving two complex problems. The one is a providing of a sufficiently powerful electric power supply without sunlight. The another is a creating of a communication channel from Jupiter to Earth with a large informational speed. Probably an additional retranslator is needed at orbit around Jupiter.

Any smaller orbit has direct Sun light. This increases temperature and noise and decreases sensitivity. Any larger orbit without Sun light at a planet-Sun L2 point has a too large period.

Or it may be mission with two observatories. The one is at the Jupiter-Sun L2 point. The another is at the Saturn-Sun L2 point. This provides accurate definitions of parallaxes immediately for a large part of directions. And for all directions is needed 1 decade for rotation of the Saturn-Jupiter line for a quarter of turn. At the same time clarifications of own movements of stars may be for a few years during this quarter of turn. And 1...2 decades are necessary for flight from Earth to Saturn.