Seriously—planetary rings are complex and complicated things.
... still here?
Okay, but don't say I didn't warn you....
Planetary Rings: General Characteristics
- Leftover material from the protoplanetary disk which was trapped within the planet's Hill Sphere, but was too gravitationally disturbed to form a larger body;
- From the debris of a larger moon pulverized by a massive impact;
- From the debris of a moon or a close-approach asteroid which came near enough to the planet to be ripped apart by tidal stresses.
* Very faint rings can also be formed from the ejecta of minor asteroid impacts on moons, or by material lost by a moon through eruptive events, such as ice geysers or sulfur plumes.
Terrestrial Planet Rings
* The first number is the top of the Earth’s thermosphere, which is where meteors occur, and therefore would be the portion of the atmosphere which would provide enough drag on ring particles to cause them to deorbit. The second number is the top of the exosphere.
Since one of the ways rings might form is through the tidal destruction of moons or small passing bodies, the Roche Limit of the body with respect to the kinds of satellites it might have is useful to know. The following table lists the Roche Limits for various Solar System bodies, should they find themselves in orbit around Earth:
Recall from this previous blog, that the Roche Limit isn't a constant, but is dependent upon the composition of the smaller body—the less an object is held together by its own gravity, the farther it has to stay from a gravity well to avoid being torn apart (unless it's held together mechanically—which is why the International Space Station isn't destroyed by gravitational tidal stresses).
For the purposes of Worldbuilding, if a terrestrial world is intended to have a ring system, its inner limit will have to be calculated taking into account the mass of the planet and the composition of its atmosphere, as well as the Roche Limit for the body which disintegrated to form the ring. In practice, the outer limit would likely be somewhere inside the orbit of the first of any major moons also in orbit around the planet. If there are no major moons, then the ring will extend out—theoretically—to the planet's Hill Sphere radius, growing more and more rarified with increasing distance from the planet. If there are small moons beyond the ring, or within it, they may act as Saturn's "shepherd" moons do, causing gaps in the ring body, with usually fairly sharp, clearly defined edges.
In general, assuming a world with an Earth-analog atmosphere, and assuming the maximum density of the ring-forming body to be equal to that of Earth, then the theoretical limits for a ring would be [1.11, 2.44] Earth radii. For lower density source objects, the outer limit will be larger, limited only by the planet’s Hill Sphere radius and the presence of any major moons also orbiting.
Ring Effects on Habitable Worlds
Depending on the density of the ring material and the axial tilt of the planet, the rings will cast a greater or lesser shadow on the planet at various times of the day and year. For ice and gas giants this may make for spectacular photography, but it has little impact beyond the aesthetic.
For rocky planets—Gaean worlds, in particular—the effects will be far more pronounced. If the ring system is stable and exists for a significant fraction of the planet’s lifetime, a ring might have climatic or other effects that would inhibit or entirely prevent the evolution of advanced life.
Assume, for example (see below), a planet with an obliquity of 30° and a ring located at [0.25, 0.5] planetary radii. (I know, those numbers are below the minimum I just stated: this is just an example for illustration purposes.) At winter solstice in the northern hemisphere, the shadow of the ring covers from 18.59° N to 60° N (see below for how to calculate these values).
For reference, Port-au-Prince, Haiti lies at 18.53° N.; Havana, Cuba is at 23.13° N.; and Atlanta, Georgia, is at 33.75° N.
Note that the ring shadow will form a straight line along the planet's surface only when it falls precisely on the equator; we have a curved object casting a shadow on another curved object. Below are three example graphics to give you an idea what really happens; a view from the south pole of the planet, from directly between the planet and its star, and from the perpendicular to a line between the planet and its star.
In the following sections, any shadow angles should be understood to refer to the point on the planet's surface which is directly "under" the planet's sun at local noon, called the sub-solar point.
The Shifting Sun Angle
Note that the angle at which a star's light strikes the surface of a planet varies throughout the planet's year, depending on the tilt of the planet's axis. In Earth's case, its obliquity of 23.439° means that the highest point the Sun reaches in the sky varies throughout the year by twice that amount: 46.878°.
For example, to an observer in Macapá, Brazil, which lies almost exactly on the equator, the Sun would pass directly overhead on either of the equinoxes, or 90° "upward" from the southern horizon (this is the sun angle, also known as the Sun's "altitude"). On the northern summer solstice in Macapá, the Sun would pass at an altitude of 113.439°, or 90° + 23.439°—in other words, 23.439˚ "into" Macapá's northern sky, or 66.561˚ "upward" from the northern horizon.
Six months later, the Sun would pass at an altitude of 66.561° or 90° - 23.439°, or 23.439˚ "short" of directly overhead. If we subtract the Sun's northernmost altitude from its southernmost altitude, we find that the total angle the Sun covers between its highest and its lowest altitudes is 113.439° - 66.561° = 46.878°, which is two times 23.439°.
For instance, for a viewer from Rockhampton, in Queensland, Australia, which is located at 23.232° S—almost precisely on the Tropic of Capricorn—the Sun passes directly overhead on the southern summer solstice (Figure 1, below), and then at 113.439° altitude on the southern autumnal equinox (Figure 2), and finally at 136.878° altitude three months later on the southern winter solstice, or only 43.122° above the northern horizon (Figure 3).
The Motion of the Ring Shadow
- Form a curved area of darkness across the planet's surface;
- Fall on different parts of the surface at different times in the year;
- "Travel" northward and southward throughout the year as the Sun's altitude changes; and,
- Vary in width depending on the the physical width of the ring and on which part of the planet's surface it is falling.
Let us imagine that the Earth has a debris ring.
The angle of latitude at the sub-solar point (where the edge of the shadow directly in line with the Sun) can be calculated from the sun angle with the following equation:
Also, the fact that the seasons are not of uniform length additionally means that the rate of travel of the shadow varies depending on:
- Whether the shadow is falling on the northern or the southern hemisphere; and,
- Whether it is traveling northward or southward.
The sun angle can also be calculated when the latitude of either north or the south edge of the shadow at the sub-solar point is known, using the equation:
𝛾 = 13.53˚
h = 2.1 Earth radii to the inner edge of the ring;
r = radius of the Earth.
Note that in the above calculations, the sun angle is relative to the ecliptic. Calculating the sun angle (or altitude) from the perspective of a viewer on the planet's surface requires extra steps.
If we graph the shadow latitudes for sun angles from 0° to 23.5° (rounding Earth's obliquity up slightly), in increments of 0.1°, we get the following:
- The shadow's latitude increases more quickly the farther north (or south) it falls, due to the curvature of the Earth. This can most clearly be seen on the green line tracking the northern (or southern) edge of the shadow, but it is discernible in the graph that the blue line is not straight, but curved slightly upward;
- The width of the shadow increases with distance from the equator until it reaches a maximum width when the northern (or southern) edge encounters the Arctic Circle. Subtracting the southern latitude from the northern latitude and multiplying by the number of miles or kilometers per degree of the planet's polar circumference gives a geographic measure for the shadow's coverage;
- The green line (the north [or south] edge of the shadow) approaches vertical at around 21.7° of sun angle. This is because the curving surface of the Earth causes the top edge of the shadow to "jump off" the surface of the Earth at the latitude of the Arctic (or Antarctic) Circle (66.561° N/S). Figure 7, above, clearly shows that at the time of the local winter solstice, the edge of the shadow passes "above"/"below" the curve of the Earth's surface. Using the second equation above, we can calculate the sun angle at which this occurs:
The figures are slightly different in the southern hemisphere, because the southern winter is the length of the northern summer.
Please note once again that all of the above assumes a ring girding the Earth, and which is 0.6 Earth radii wide, extending between 2.1 and 2.7 Earth radii above the surface. A differently sized ring around a differently sized planet with a different axial tilt would produce entirely different results.
The latitude at which the ring shadow encounters the terminator can be loosely estimated by the following equation:
Let α = 11.7195, d = 2.1 planetary radii for the inner edge of the ring, and dʹ = 2.7 planetary radii for the outer edge of the ring.
The reason this is only an estimate is because it assumes the shadow falling from the local zenith point. In fact, the shadow striking the terminator falls from a point along the ring to the east and west of the zenith by the radius of the planet. Due to the curvature of the ring, this point actually lies “below” the level of the zenith point on the edge of the ring. Also, again, bear in mind that the shadow will not follow a straight path between these latitudes, but will curve along the surface of the planet.
Nevertheless, if we select the location on the planet’s surface where the outer edge shadow meets the terminator, and rotate the planet back one quarter of a rotation to local noon, we can see that the specified location does not experience the shadow at any time during its daylight on that particular rotation, because the latitude of the outer edge of the ring at local noon is 12.523˚ farther south. The ring would be visible in the sky, but the local star would be “above” it until right at the moment of sunset.
Conversely, a sub-solar point which is directly under the shadow of the outer edge at local noon (latitude 21.54˚) will rotate out of the shadow at some time during the day, because the shadow of the inner edge of the ring falls at a higher latitude (25.826˚) on the terminator than it does at local noon.
The times of day at which a given location rotates into and out of the shadow will be dependent upon a number of factors:
- The time of year: is the shadow falling on the surface at the time of year in question?
- The sun angle: how far above or below the equator is the shadow falling? Also, what is the width of the shadow on the ground?
- The latitude of the location: is the location already in shadow when the sun rises, or does it move into the shadow at some point between dawn and noon?
As you can see, having an inhabited planet with a ring is a fascinating idea, but describing life there with scientific and mathematical accuracy is a complex undertaking.
Sorry you asked?