Mercury has no atmosphere, just an exosphere. The highest mountain range on Mercury are the Caloris Montes, reaching up to 1.9 miles (3 km). What if a spacecraft tried to orbit Mercury at an altitude of 3 miles (4.8 km)? Since Mercury has no considerable atmosphere, would the spacecraft have a stable orbit? If no, why not? A similar question could be asked on the Moon which however wouldn't be a good example due to the extreme mascons of the Moon. Such a question could however work with Ceres and any other spherical celestial body without a considerable atmosphere too probably.
On May 6, 2016 NASA’s MESSENGER mission – which orbited Mercury from 2011 until 2015 – unveiled the first global digital elevation model, showing the topography, or highs and lows of natural features, across the entire innermost planet.
This new model reveals a variety of interesting topographic features, as shown in the animation above, including Mercury’s highest and lowest points. The highest point on Mercury is at 2.78 miles (4.48 km) above Mercury’s average elevation, located just south of the equator in some of Mercury’s oldest terrain. The lowest elevation, at 3.34 miles (5.38 km) below Mercury’s average. It’s found on the floor of Rachmaninoff basin, an intriguing double-ring impact basin suspected to host some of Mercury’s most recent volcanic deposits.
So from a topographical point of view at least a few orbits at 3 miles without colliding a surface feature.
The question now asks
Could a spacecraft make two stable orbits around Mercury at an altitude of three miles?
partly because of my comment
questions about an orbit being stable or not stable are often ambiguous because there are always several things that affect any orbit, but most of them are pretty small and the effects take a long time to make significant changes. Instead of asking "Is or isn't stable" it's probably better to ask *Roughly how long could a spacecraft in a low Mercury orbit at an altitude of 3 km remain in a low Mercury orbit without hitting the surface or drifting away?" Or you could define "stable" as lasting for perhaps ten years arbitrarily.
What's your definition of stable? Would you consider 1 orbit stable? Or, say 100?
but I'm going to add a supplemental answer to address long-temp stability.
The paper in Journal of Geophysical Research The gravity field, orientation, and ephemeris of Mercuryfrom MESSENGER observations after three years in orbit (also available here) describes a detailed analysis of Mercury's gravity field.
We have analyzed 3 years of radio tracking data from the MESSENGER spacecraft in orbit around Mercury and determined the gravity field, planetary orientation, and ephemeris of the innermost planet. With improvements in spatial coverage, force modeling, and data weighting, we refined an earlier global gravity field both in quality and resolution, and we present here a spherical harmonic solution to degree and order 50. In this field, termed HgM005, uncertainties in low-degree coefficients are reduced by an order of magnitude relative to earlier global fields, and we obtained a preliminary value of the tidal Love number k2 of 0.451±0.014. We also estimated Mercury’s pole position, and we obtained an obliquity value of 2.06±0.16 arcmin, in good agreement with analysis of Earth-based radar observations. From our updated rotation period (58.646146 ± 0.000011 days) and Mercury ephemeris, we verified experimentally the planet’s 3 : 2 spin-orbit resonance to greater accuracy than previously possible. We present a detailed analysis of the HgM005 covariance matrix, and we describe some near-circular frozen orbits around Mercury that could be advantageous for future exploration.
When we hear about "frozen orbits" we normally think of some special orbits around the Moon that are "safer" and more "stable" than most, orbits where you could insert a spacecraft and count on it still being in orbit a few months or perhaps years later without needing propulsive station keeping.
- Orbital Mechanics with Numerit; Frozen Orbit Design
- Frozen Orbits About the Moon
- Bizarre Lunar Orbits
The paper about Mercury presents solutions for frozen orbits for altitudes of 300, 500 and 1000 kilometers about a reference radius of 2440 km.
Near an inclination of 60 degrees all three altitudes considered seem to have near-circular frozen orbit solutions, so it's possible lower altitudes could have at least short term solutions without impacting the surface.
However note that this calculation uses an order fifty gravity model and "(o)nly the zonal gravity coefficients of HgM005 were considered" so the closer you get to the surface the more that higher order (fine lumpiness) terms would have to be considered, as well as the fact that HgM005 may not be 100% correct.
This doesn't address at all two orbits at three miles directly, but it suggests that a proper answer would require a pretty accurate high-order gravity model to cut things so close.
The image caption provides an intuitive preview:
Figure 10. Computed eccentricity for frozen orbits of varied inclination and semimajor axis (referenced to R = 2440 km to yield altitude h). Filled areas indicate that the frozen eccentricity is above the maximum acceptable eccentricity (dash-dotted line) and would lead to surface impact. Dashed lines indicate that the argument of pericenter 𝜔 is 90◦, whereas solid lines indicate 𝜔 = 270◦ (pericenter over the southern hemisphere). Only the zonal gravity coefficients of HgM005 were considered.