Hi there! Today I will present you a new way to weigh an inner satellite of a giant planet. This is the opportunity for me to present you Weighing Uranus’ moon Cressida with the η Ring by Robert O. Chancia, Matthew M. Hedman & Richard G. French. This study has recently been accepted for publication in The Astronomical Journal.

The inner system of Uranus

Uranus is known to be the third planet of the Solar System by its radius, the 4th by its mass, and the 7th by its distance to the Sun. It is also known to be highly tilted, its polar axis almost being in its orbital plane. You may also know that it has 5 major satellites (Ariel, Umbriel, Titania, Oberon, and Miranda), and that it has been visited by the spacecraft Voyager 2 in 1986. But here, we are interested in its inner system. If we traveled from the center of Uranus to the orbit of the innermost of its major satellites, i.e. Miranda, we would encounter:

• At 25,559 km: the location where the atmosphere reaches the pressure 1 bar. This is considered to be the radius of the planet.
• Between 37,850 and 41,350 km: the ζ Ring,
• At 41,837 km: the 6 Ring,
• At 42,234 km: the 5 Ring,
• At 42,570 km: the 4 Ring,
• At 44,718 km: the α Ring,
• At 45,661 km: the β Ring,
• At 47,175 km: the η Ring,
• At 47,627 km: the γ Ring,
• At 48,300 km: the δ Ring,
• At 49,770 km: the satellite Cordelia (radius: 20 km),
• At 50,023 km: the λ Ring,
• At 51,149 km: the ε Ring
• At 53,790 km: the satellite Ophelia (radius: 22 km),
• At 59,170 km: the satellite Bianca (radius: 26 km),
• At 61,780 km: the satellite Cressida (radius: 40 km),
• At 62,680 km: the satellite Desdemona (radius: 34 km),
• At 64,350 km: the satellite Juliet (radius: 47 km),
• At 66,090 km: the satellite Portia (radius: 68 km),
• Between 66,100 and 69,900 km: the ν Ring,
• At 69,940 km: the satellite Rosalind (radius: 36 km),
• At 74,800 km: the satellite Cupid (radius: 9 km),
• At 75,260 km: the satellite Belinda (radius: 45 km),
• At 76,400 km: the satellite Perdita (radius: 15 km),
• At 86,010 km: the satellite Puck (radius: 81 km),
• Between 86,000 and 103,000 km: the μ Ring,
• In the μ Ring, at 97,700 km: the satellite Mab (radius: 13 km)
• At 129,390 km: the satellite Miranda (radius: 236 km).

The rings of Uranus are being discovered since 1977. Originally it was from star occultations observed from the Earth. Then Voyager 2 visited Uranus in 1986, which revealed other rings, and more recently the Hubble Space Telescope imaged some of them, permitting other discoveries.. Most of them have a width of ≈1 km.
All of the inner moons have been discovered on Voyager 2 images, except Cupid and Mab, which have been discovered in 2003, once more thanks to Hubble. On the contrary, the major moons have been discovered between 1787 and 1948.

Today we will focus only on

• At 47,175 km: the η Ring,
• At 61,780 km: the satellite Cressida (radius: 40 km).

The η Ring is very close to the 3:2 mean-motion resonance (MMR) with Cressida, which means that any particle of the η Ring makes 3 revolutions around Uranus while Cressida makes 2. As a consequence, Cressida has a strong gravitational action on the η Ring.

Gravitational interactions

How do we know the mass of planetary bodies? When we send a spacecraft close enough, the spacecraft is deviated, and from the deviation we have the gravity field, or at least the mass. If we cannot send a spacecraft, then we can invert, i.e. analyze, the interactions between different bodies. We know the mass of the Sun thanks to the orbits of the planets, we know the mass of Jupiter thanks to the orbits of its satellites, and the deviations of the spacecraft. We can also use MMR. For instance, in the system of Saturn, the mass ratios between Mimas and Tethys, between Enceladus and Dione, and between Janus and Epimetheus, were accurately known before the arrival of Cassini, thanks to resonant relations.

We can have resonant interactions between a satellite and a ring, as well. A ring is actually a cloud of small particles, and the way their motion is affected reveals the gravitational interaction with something. When you have a MMR, then the ring exhibits streamlines, which give a pattern with equally spaced corners. From the number of these corners you can determine the MMR involved, and from the size of the pattern you get the mass of the disturbing satellite. This is exactly what happens here, i.e. 3:2 MMR with Cressida affects the η Ring in such a way that you can read the mass of Cressida from the shape of this ring. But for that, you need to be accurate enough on the location of the ring.

The data

The authors used 49 observations, including 3 Voyager 2 ones, the other ones being star occultations by rings. Such an observation should be anticipated, i.e. the relative position of Uranus with respect to thousands of stars is calculated, then the star has to be observed where possible, i.e. in a place where it will be high enough in the sky, and of course at night. You measure the light flux coming from the star, which should be pretty constant… and is not because of the variability of the atmospheric thickness since the star is moving in the sky (remember: the Earth rotates in one day), so you have to compensate with other stars… and if you detect a flux drop, then this means that something is occulting the star. Possibly a ring.
Most of the observations were made in the K band, i.e. at an infrared wavelength of 2.2 μm, where Uranus is fainter than its rings. These observations have been made between 1977 and 1996. Since then, the opening of the rings is too small, i.e. we see Uranus by the edge, which reduces the chances to occult a star.

Methodology

The authors made a least-square fit. This means that they fitted their corpus of observations with a shape of the ring as R-A cos (mθ), where R is a constant radius, A is an amplitude of distortion of the ring, θ is the angle (a longitude), and m is a factor giving the frequency of the distortion, which could be related to its cause, i.e. the orbital motion of the satellite affecting the ring. You fit R, A and m, i.e. you adjust them so as to reduce the difference (the error, which is mathematically seen as a distance) between your model and the observations. From R you have a ring (and you can check whether there should be a ring there), from A you have the mass of the satellite, and from m and have its frequency (and you can check whether a known satellite has this frequency).
The authors show that the highest effect of the inner satellites on the rings should be the effect of Cressida on the η Ring, thanks to the 3:2 MMR.

Results

The authors find that Cressida should have a density of 0.86±0.16 g.cm^-3, which is lighter than water. Usually these bodies are supposed to be kind of porous dirty ice, which would mean this kind of density. This is the first measurement of the density of an inner satellite of Uranus. A comparison with other systems shows that this is much denser than the inner satellites of Saturn. However, the inner satellite of Jupiter Amalthea has a pretty similar density.

Finally the authors say that they used this method on other rings, and that additional results should be expected, so we stay tuned. They also say that a spacecraft orbiting Uranus would help knowing these satellites. I cannot agree more. Some years ago, a space mission named Uranus Pathfinder has been proposed to ESA, and another one, named Uranus orbiter and probe, has been proposed to NASA.

The study and the authors

• The study, made freely available by the authors on arXiv, thanks to them for sharing!
• The profile of Robert Ormal Chancia on ResearchGate,
• The web page of Matt Hedman,
• The one of Richard French.

That’s it for today! Please do not forget to comment. You can also subscribe to the RSS feed, and follow me on Twitter and Facebook.