TESS orbit and moon resonance

Question

On April 17, SpaceX will launch TESS satellite.

Satellite description:

The Transiting Exoplanet Survey Satellite (TESS) is a planned telescope for NASA'S explorer progra. designed to search for exoplanets using the transit method in an area 400 times larger than that covered by the kepler mission.

The wiki article talks about the orbit and to quote

In order to obtain unobstructed imagery of both the northern and southern hemispheres of the sky, TESS will utilize a 2:1 lunar resonant orbit called P/2, an orbit that has never been used before (although IBEX uses a similar P/3 orbit). The highly elliptical orbit has a 373,000 km (232,000 mi) apogee, timed to be positioned approximately 90° away from the position of the Moon to minimize its destabilizing effect. This orbit should remain stable for decades, and will keep TESS's cameras in a stable temperature range. 

Does name P/2 relate to orbital resonance ratio? So P/3 means 1:3 resonance?

Also, why it has 90 degree offset and not 180 degrees ? Is the optimal offset derivable for minimal destabilizing effect of moon by some simple math ?

Answer 1

An excellent review of all of the considerations and work that has gone into designing TESS' orbit can be read in the ArXiv preprint A High Earth, Lunar Resonant Orbit for Lower Cost Space Science Missions and I'll draw primarily from this, and the excellent YouTube video Transiting Exoplanet Survey Satellite (TESS).

The development of TESS' remarkable orbit and scientists who worked on this project are also discussed in the 2013 NASA article New Explorer Mission Chooses the ‘Just-Right’ Orbit:

Principal Investigator George Ricker likes to call it the "Goldilocks orbit" — it’s not too close to Earth and her Moon, and it’s not too far. In fact, it’s just right.

above: Goddard engineers Chad Mendelsohn, Trevor Williams, and Don Dichmann helped formulate NASA’s next Explorer mission’s never-before-used orbit. Cropped. Credits: NASA Goddard/Pat Izzo. From here.

---

From the ArXiv paper:

Orbital perturbations from Earth oblateness, the Sun, and the Moon could in principle make the mission orbit unstable, leading to a violation of mission constraints. The mission-orbit perigee may drop below the GEO belt after a few years, or the inclination may drop low enough to create mission-ending eclipses. The long-term stability of the mission orbit depends on its initial orbit elements, which are driven in turn by the transfer orbit.

The TESS P/2-HEO mission orbit must satisfy two requirements to ensure a feasible mission: 1) all eclipses must be less than 6 hours, and 2) perigee must remain between 7 and 22 RE throughout the 4-year mission.

Requirements of TESS' orbit design include:

1. Stable orbit for years or preferably decades so spacecraft does not require large amounts of on-orbit propulsion cost, weight, and complexity. Look how long Hubble has lasted; you'd like an orbit that has the potential to be useful for decades.
2. Avoiding long period eclipses of the Sun that could lead to power loss with limited battery life.
3. Sufficiently close Earth periapsis to allow for reasonably short and infrequent interruptions from observation for high volumes of data transfer using a modest antenna size and power; less than 6 hours with periapsis between 7 and 22 Earth radii throughout the 4-year mission.
4. Avoid "You can't get there from here" scenarios (from Bert and I, also here and here). In other words, there must be a practical way to reach this orbit using a reasonable, affordable launch vehicle and a lunar swing-by maneuver.

All of these conditions have been met by TESS' final orbit following the procedure described in the ArXiv preprint.

Had 0 degrees been chosen rather than 90 degrees, the orbit would have been quickly perturbed beyond usefulness as the distance of closest approach between TESS and the Moon would be small, producing a strong, asymmetric, and resonant "tug".

The small distance of TESS' closest approach to the orbital path of the Moon and therefore the necessity of the 90 degree phasing is easily understood. For efficient and fast data transmission to Earth, periapsis must be very small, say 10 Earth radii or about 0.2 times the Moon's semi-major axis. An orbit with 0.5 the Moon's period would have a semi-major axis of 0.5^(2/3) or about 0.63 times the Moon's semi-major axis. Apoapsis is twice the semi-major axis minus periapsis, or 2 times 0.63 minus 0.2 or 1.06 times the Moon's average distance from the Earth.

Even with this carefully engineered orbit, you can see that the near cancellation of the Moon's gravitational perturbation on average still results in quite a long-term dance of the orbit shown here in a synodic frame rotating with the Moon's period.

---

below: Cropped from the video Transiting Exoplanet Survey Satellite (TESS) at 06:46. A lunar swing-by, followed by a "period-adjust maneuver at PLEP" at periapsis finalizes TESS' orbit with the proper shape and inclination plus phasing it so that the Moon's position will be at 90 degrees to either side at apoapsis.

below: Cropped from the video Transiting Exoplanet Survey Satellite (TESS) at 06:57. The Moon's position will be at 90 degrees to either side at apoapsis, here shown on one side, the Moon would be near the top of the image after one 13.7 day full period of TESS' orbit. If the resonant orbit were phased at 0 degrees, the short distance of closest approach would quickly destabilize TESS' orbit.

Answer 2

Yes, P/2 means half the lunar period, and P/3 means a third. There's a discussion of the new IBEX P/3 orbit, including a cool image, here.

The 90 degrees is a requirement because it stays away from the Moon.

Consider the orbit of the moon as a circle labelled as a clock. The moon goes from 12 to 3 to 6 to 9 to 12 in one period. In a half a period, roughly 14 days, it goes from 12 to 6.

Start with the Moon at 12, and TESS 90 degrees away at 3. A quarter period (half of TESS's orbit), TESS is in the center and the Moon is at 3. Another, TESS is out at 3 and the Moon has moved on to 6. They never get close to each other.

If they were 180 degrees out on one orbit, a half-lunar-orbit later (a full TESS orbit later), they'd be in the same place. Not good.