The Webb at a million miles, really?

George Rebane

The recently launched Webb space telescope will be able to peer further back in time and see more of what was then there.  As an amateur cosmologist, I’ve been interested in this new instrument for some years now.  It brings these new capabilities to bear because of its very large mirror and that its imaging plane is designed to be sensitive to infra-red light.  From high school we recall that IR has a long wavelength, and therefore can penetrate through all kinds of dust, water vapor, and other tiny stuff in space.  It’ll be able to make out far-off galaxies through dust clouds that formed around 13.5B years ago, only 300M years after the Big Bang.  (more here)

In the media we have been told that the Webb will orbit Earth’s L2 point shown in the above figure which also indicates the location of the other four Lagrange points.  The L2 point is on the straight line connecting Earth and the Sun, and reported to be about a million miles beyond Earth’s orbit.  Since this point in space stays on such a straight line, the angular rotation of L2 therefore equals that of Earth; along with our planet, it completes one rotation of the Sun in 365.25 days.

The Lagrange points are unique locations in a three-body configuration where the gravitational attraction of the two larger bodies provide ‘stable’ orbital points for much lighter third bodies.  Of the three kinds of stability – conditional (marble on large ball), unconditional (marble in a bowl), neutral (marble on a flat plane) – only L4 and L5 are unconditionally stable in that if a lighter body there is perturbed, it will remain in the L4 or L5 region (cf. The Travelers of Jupiter).  The other three points are conditionally stable, a perturbation from, say, the solar wind or the gravitational effects of the other planets, will push the lighter body away on a trajectory never to return.  Hence, to keep the Webb telescope near L2 requires little accurately directed squirts of thrust now and then.

Ever since I was a physics major, I was always intrigued by the balance of forces that gave rise to the Lagrange points, but having solved a number of other multi-force problems in my studies, I never took the time to actually calculate and confirm the existence of these counter-intuitive stable points in orbital mechanics.  But the launch of the Webb got me off my duff, and I decided to take a closer look and see what was required for the L2 to exist and how far was it really from Earth – a “million miles” sounded a bit too coincidental.

So, from the figure, let the Earth-to-L2 distance be r_EL2.  For any kind of stability, at the L2 point we must have its centrifugal force outward equal the sum of the Sun’s and Earth’s gravitational forces inward (toward the Sun and the Earth).  We can compute the actual value of r_EL2 from some straightforward formulas, namely Newton’s gravitational attraction and the centrifugal force formulas known from physics.  I show this derivation in the panel below.

But first we need to define the terms.  The m_x are the masses for x = Sun, Earth, Webb at L2.  The r_x are the respective distances in the three-body orbital plane.  G is the gravitational constant in Newton’s law of gravity, and τ_E is Earth’s orbital period (one year) around the sun.

These calculations, along with the proper values for all the terms, are shown in the spreadsheet figure below.  The astute spreadsheet driver will note that I used Excel’s nonlinear gradient search feature in its ‘Solver’ function to solve for the value of r_EL2 that drives the equation in the box to zero – i.e. the left-hand side is the ‘utility’ to be minimized by the correct value of r_EL2; its minimized value is highlighted in yellow.  (This saved me from having to produce the analytical solution for a cubic equation – discretion is the better part of valor.)

And there you have it.  The Earth-L2 distance is actually 943,769 miles, and not the nice rounded one million miles – a 6% error.  And BTW, my calculated values agree with what’s published in the literature, so there.  Newton still rules for these kinds of calcs.

Comments

[scenes]

I must say that I was surprised there was such a thing as an L2, but I guess that an L4 & L5 implies 1,2,3. I believe that Webb has an orbit around it.

[George Rebane]

scenes 1223pm - Webb will not have what one could consider a regular orbit around L2, but instead be so guided by ad hoc thrusts so as to remain in L2's vicinity. I didn't want to put too fine a point on it, but due to the many influences that dynamically distort Earth's gravitational field, L2 is not a relatively stationary point, but wanders around a bit - e.g. every time the Moon swings by in its orbit around the Earth. This requires that Webb also has to 'chase' it to remain in the L2 vicinity.

[scenes]

aah, thanks. I didn't pick that up from the articles about it. It's always presented in a more simplified fashion. Is the wandering predictable?

[George Rebane]

scenes 216pm - The wandering is predictable for relatively short future horizons. It's an estimation problem (see Kalman filters) that is affected by the error in the Webb's current state (position, velocities, and roll rates) and the accuracy of the implemented control (direction, timing, and duration of correction thrusts), all of that is combined with the prediction of the multi-body (more than three) gravitational field dynamics at L2. And if that isn't enough, then add to it the total unpredictability of the solar wind which does have an impact on Webb because of its huge sun-shade that is kept orthogonal to the sun at all times. The resulting correction trajectory is at best what technically is called chaotic. A lot of the correction calculations and position keeping can be done onboard, thereby eliminating some long distance errors to Earth. The whole control regime is continually recomputed and optimized to require a minimum expenditure of onboard fuel, since that is what we're told determines the useful life of Webb. My own feeling is that if the damn thing works half as good as its specs, then they'll make a refueling run out there when needed. No one knows how many even better space telescopes will be put into Lagrange points in the next 20-30 years. And less fuel will be needed at L4 and L5.

[The Estonian Fox]

George, you just HAD to do it. Couldn't just accept 1,000,000 miles because it seemed too contrived! Tony Fauci said, and I quote, "What's 56,230.62 miles between friends?" And he is the SCIENCE you know.

Anyway, good job. I've always known them as the Trojans, the captured bodies at L4-L5 around Jupiter. And Saturn doesn't seem to have many Trojans, if any, most likely because Jupiter & Saturn are sort of resonant-locked in their nearly 5:2 ratio of orbital periods. Jupiter is just too massive to allow any stray bodies 'near' Saturn, even 60 degrees away.

I just hope that the Webb doesn't have a problem similar to the one experienced by Hubble. I still have my 1986 copy of Discover magazine, which had a long article about the extensive testing/quality control procedures at Perkin-Elmer, to ensure clear-seeing in the Hubble. And we all got to literally 'see' how well that turned out at first launch.

I agree with you that if the Webb performs as expected, they will indeed make refueling (Tesla self-driving robotic??) runs to it. Don't forget, Elon already has a Tesla out there somewhere - who's to say that he didn't load the auto-driver software onto it? Just need to attach a small light-sail, and away we go.