Living on the habitable moon of a large planet is a common subject of fascination in science fiction, particular visual art, largely for the simple reason that it allows you to stick that planet in the sky and immediately communicate to the audience that we're in a weird scifi place and they should expect weird scifi things to happen.But the influence of a planet on its moons can go beyond a pretty skybox. Tidal interactions and radiation could influence the habitability of moons, but today I want to specifically focus on how a planet can influence the patterns of light and dark across a moon's orbit, both through eclipses when the planet blocks light from reaching the moon and reflected light bounced from the planet onto the moon. I sometimes see people concerned that these may be significant hazards for habitability of these moons, but in almost all cases this seems to be unlikely; if a planet orbits in the habitable zone, there really isn't any strong reason that a moon of that planet couldn't be habitable. But the influence can still be quite significant in terms of how it feels to live on the surface of that moon and could conceivably factor into the life or cultures that evolve there.
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First, an important point to establish: almost all moons within the solar system are tidal-locked to their planet, meaning that the planet rotates at the same rate as it orbits the planet such that one hemisphere permanently faces the planet and the other permanently faces away, and we can likely expect much the same will be true elsewhere. Tidal forces vary strongly with distance, and the distances between planets and moons tend to be much smaller than those between stars and planets, such that even if they initially form with very fast rotation, moons should tend to tidal-lock to their planets within millions of years, if not even faster. Perhaps a very distant moon at the edges of orbital stability could hold out for longer without tidal-locking, even billions of years, but such a moon would also be far enough from its planet to see little influence on its lighting conditions, so that's not really a case we need to worry about here. The influence of another large nearby moon could also cause chaotic rotation with no fixed axis of rotation, as is the case for Hyperion and likely the minor moons of Pluto, but again these are edge cases that are likely to be rare for large, habitable moons; they may be interesting cases to investigate in their own right, but I'll keep to more standard cases here.
Remaining tidal-locked to the planet as it orbits around means that, relative to the star, the moon rotates around and experiences a full day and night roughly once every orbit; in practice, because the planet moves along its orbit at the same time and so the direction to the star shifts, the moon has to complete either slightly more than a complete orbit (if it orbits in the same orientation the planet does, the more typical case) or slightly less (if it orbits in reverse to the planet) for the star to appear to return to the same position in its sky; Earth's moon, for example, completes an orbit about every 27 days but has a solar day lasting a bit over 29 Earth days (and this latter figure is also the length of the lunar phase cycle as seen from Earth). But usually this is a fairly minor discrepancy save for again some edge cases where the moon orbits close to the limits of stability.
So far as I can tell, it shouldn't be possible
for a moon to be tidal-locked to the planet's star rather than the
planet itself; the constraints on stable orbits of the planet and the
strength of tidal forces are determined by some of the same factors in
such a way that ensures that any moon orbiting a planet can always
expect to receive stronger tidal forces from the planet than from the
star. So the peculiar conditions and climate of a tidal-locked planet,
with permanent light for one hemisphere and permanent dark for the
other, aren't relevant here. Strong gravitational influence from the star may cause rapid orbital migration of the moon, limiting the time it can remain in the planet's orbit before it inevitably either passes below its Roche limit and breaks apart or escapes into its own orbit of the star, but that's mostly an issue for planets of relatively small stars; habitable moons of planets orbiting a sunlike star should be fine.
Being tidal-locked to the planet also means that the planet appears to remain static in the moon's sky, and only half of the moon's surface will ever see the planet at all—a bit more including some areas that might see at least a sliver of the planet's face above the horizon. A moon with some orbital eccentricity and axial tilt could also experience some libration, a slight shift in its orientation relative to the planet such that the planet does appear to move around a little in the sky and some areas near the edge of the planet-facing hemisphere will see it rise up just past the horizon and sink back down. These combined effects allow for about 59% of our moon's surface to see some part of Earth at least occasionally. A moon of a large gas giant would likely feel stronger tidal forces and so have very little libration, but the planet may appear much larger in the moon's sky, such that its edges remain visible farther past the edges of the planet-facing hemisphere.
For the hemisphere of the moon facing away from the planet, their sky would appear much the same as that of a planet in an equivalent orbit with the same rotation period, with a fairly pedestrian cycle of day and night. If the planet had other moons, their motion through the sky might appear peculiar, but I'm going to avoid getting sidetracked into a discussion of the potential nature of early astronomy on such a world. I will say that I've personally always quite liked the idea of a society developing on the outward-facing hemisphere of a moon and then sending an expedition towards the planet-facing hemisphere, who are surprised to discover an immense orb apparently just floating in the sky.
At any rate, for this post I'm concerned with conditions on the planet-facing hemisphere, with a particular focus on the regions near the center of that hemisphere, with the planet directly overhead.
Eclipses
With all that preamble out of the way, the first important factor is eclipses: As a moon orbits its planet, it will sometimes pass through its shadow, blocking all or at least part of the light to the moon. The nature of these eclipses depends in large part on the apparent diameter of both the planet and the star in the moon's sky, which can be calculated like so:
Ī“ = apparent diameter
r = radius of object (any unit so long as D is the same)
D = distance from observer to center of object
The Sun and Moon both have an apparent diameter of about 0.53° in Earth's sky, but gas giants can be much larger in their moons' skies, as big as 67° for Jupiter as seen from its innermost moon Metis, but that's an extreme case; most large moons of the gas giants in our solar system see the planet as about 5-15° wide in the sky. Note that if the planet has an apparent diameter of 10°, then, walking along the moon's equator, you should be able to move about 5° past the edge of the planet-facing hemisphere before it completely disappears below the horizon.
For a moon to experience a total eclipse, its planet must have a larger apparent diameter than the star, such that it can block the star's full disk as seen from the moon. In that case, the maximum possible length of a total eclipse will depend on their relative sizes and the length of the moon's day:
te = maximum length of total eclipse (any unit so long as te is the same)
td = length of full day and night
Ī“p = apparent diameter of planet
Ī“s = apparent diameter of star
(adding the apparent diameters instead will give the maximum length of at least partial eclipse)
In essence, from the moon's perspective the star completes one full circuit of 360° through the sky once every day, moving at about a constant rate; if the planet is static in the sky and perfectly positioned to block the star for as long as possible, the total eclipse will last from the point the last of the sun's disk passes behind the planet to the point when the first part of the sun's disk peeks out on the other side, and the angle between the star's apparent position at those points compared to a full circle corresponds to the portion of the moon's day occupied by the eclipse.
- If the moon orbits on the exact same plane as the planet orbits the star, then it will reliably experience an eclipse every day, always of this same maximum length.
- If the moon's orbit has some nonzero inclination relative to the planet's orbit, but still lower than half the apparent diameter of the planet minus half the apparent diameter of the star, it will have a total eclipse every day, but varying in length, often shorter than this maximum.
- If the inclination is higher but still less than half the planet's apparent diameter plus the star's full apparent diameter, it will have at least a partial eclipse every day, but not always a full eclipse.
- At higher inclinations, there will be two periods in the year with no eclipses at all and 2 periods where eclipses may happen, as the shifting orientation of star, planet, and the moon's orbit causes its orbital path to move in and out of the planet's shadow. The higher the inclination, the longer the non-eclipsing periods are relative to the eclipsing periods; if the eclipsing periods are each shorter than one orbit of the moon, they may at least sometimes pass without an eclipse occurring at all.
Given a high inclination and long orbit, a moon might pass whole years without eclipsing; but, unless there's some exact resonance between the moon's orbit and planet's year (which so far as I can tell is unlikely), there will always be at least occasional total eclipses so long as the planet's apparent diameter is large enough.
In some edge cases these patterns could also vary a bit between latitudes of the moon; to the perspective of the moon's north pole, the planet's disk in the sky will be offset slightly to the south relative to how it appears at the moon's equator, so there may be times when the sun just appears visible past the edge of the planet at the pole while it's blocked at the equator; put another way, the planet's shadow (or more specifically the umbra) may only pass over part of the moon.
The exact length of any individual eclipse depends on complex trigonometry I won't get into here. But even going by that maximum eclipse length, that 5-15° value for the typical apparent size of our system's gas giants from their large moons implies that eclipses usually occupy only a small portion of the day—though it bears emphasizing that this is a portion of the full day and night, not just daylight. So in the extreme, areas on the Jupiter-facing side of Metis may lose as much as 1/3 of their potential daylight hours to eclipses. This might have some impact on the climate of a larger world with an atmosphere, but we'll come back to that later.
Near the middle of the moon's planet-facing hemisphere, eclipses would occur at around midday, plunging the surface into darkness for a time between morning and afternoon periods of light. Near the eastern side of the planet-facing hemisphere (taking east as the direction the moon rotates, the same as Earth), eclipses would occur late in the day, and towards the western side, they'd occur in the morning; moving north or south would make little difference in eclipse timing, though it might subtly affect its length in ways that again would take tricky trigonometry to work out. Near the very edges of the planet-facing hemisphere would be areas where the planet's position in the sky meets the horizon such that there is no gap between the eclipse and sunrise or sunset, so it effectively just somewhat increases the length of night. There may also be areas where the planet appears to just touch the horizon, with gaps between the planet and horizon towards the edges; depending on the exact positioning, one might catch brief glimpses of the sun as it passes from below the horizon to behind the planet, or if the moon's orbit is slightly inclined there might be periods where this happens and others where it remains obscured.
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| Potential morning patterns at the western edge of the planet-facing hemisphere, where the planet appears to touch or almost touch the horizon; at the eastern edge, these could occur in reverse in the evening. |
To be clear, for cases where the planet is much larger than the moon, the eclipse is close to simultaneous across the moon (it begins and ends a bit sooner in the west, but you can expect a long period where the whole planet is shaded), this just reflects different parts of the planet-facing hemisphere being in different parts of their day when it happens.
Reflected Light
The other major factor to consider here is reflected light. When the lit side of the planet is visible from the moon, light from the star will be reflected off the planet's surface and hit the moon, illuminating it. Even in the most extreme cases this light will be substantially less than the moon receives directly from the star by day, but the planet-facing side of the moon will be facing the lit side of the planet mostly during its own night, when that reflected light will make more of a difference. On Earth, reflected light from our moon can substantially improve visibility at night despite being negligible by day, and that's just from an object half a degree wide in the sky that, despite appearances, is actually quite dark, reflecting less than 1/7 the light that hits it.
Ev = illuminance of moon surface (lx)
z = planet zenith angle (degrees)
L = star luminosity (solar luminosities)
p = planet geometric albedo
r = planet radius (Earth radii)
Ī± = phase angle of moon's position from planet's lit side (degrees)
Ds = distance from star to planet (AU)
Do = distance from observer on moon's surface to center of planet (AU)
A few important notes about these parameters:
- The important measure of albedo here is geometric albedo, which is different from the Bond albedo we usually use when discussing a planet's temperature and climate. Bond albedo measures the portion of all light reaching the planet that is reflected away in any direction, while geometric albedo measures how much light is reflected back towards the light source, relative to what would be reflected by an idealized diffusely reflecting flat disk (and if we make some simplifying assumptions about the nature of a planet's surface, this gives us a sense of how much is reflected in other directions as well). These two values are generally correlated but not the same; Earth has a Bond albedo of about 0.3 but geometric albedo of 0.43.
The solar system's gas giants have geometric albedos around 0.5, but for a habitable-zone gas giant this might differ depending on the dominant type of cloud cover: we might expect the surface to be dominated by water clouds, giving them a bluish-white exterior with quite high albedo, but some research suggests they might form a sulfur haze above these clouds, tinting them more orange and lowering their albedo somewhat. As another point of reference, Venus, coated in white sulfuric acid clouds, has a geometric albedo of 0.69. Cloud cover and color could also conceivably vary with latitude, but let's not complicate matters more than we have to.
- The phase angle measures the angle at which the planet is viewed relative to looking directly at the planet's lit side, from 0° to 180°. For a moon on a circular orbit with zero inclination relative to the planet's orbit, this neatly corresponds to the phase of its orbit relative to the point when it's directly between the planet and star, but with the quirk that it counts down as the moon moves from behind the planet to this point and then up as it moves back around the planet: A phase angle of 0° shows the entire lit hemisphere of the planet, with the moon again directly between planet and star; 90° shows half the lit face of the planet, with the planet-moon axis perpendicular to the planet-star axis; and 180° shows only the unlit hemisphere of the planet, with the moon directly across the planet from the star, and this in the middle of its eclipse. For observing the moon from Earth, these correspond (roughly) to full moon, quarter moon, and new moon (though for observing Earth from the moon, it's reversed with 0° phase angle at new moon). If the moon's orbit is inclined, the relationship between orbital phase and viewing phase angle is somewhat more complicated, with the moon not always reaching 0° or 180° phase angle on every orbit.
Close to 0° phase angle, it's possible that the moon may shade part of the planet's surface, reducing the amount of light that reaches the planet and thus the amount that reflects back on the moon. For a small moon of a much larger planet, this probably makes little difference—though it might have some intriguing implications for astronomy on the moon—but as we'll see later there are some cases where it could be more impactful.
- The planet's zenith angle is the angle from its center position to the top of the sky, from 0° for a planet directly overhead at the center of the planet-facing hemisphere to 90° for a planet sitting on the horizon at the edge of that hemisphere. Note that this corresponds directly to the angle on the planet's surface from the center of the planet-facing hemisphere, e.g. degrees west or east from that point along the equator. As the planet gets lower in the sky, illuminance will decrease much as light from the sun decreases towards evening—though because the planet has width and isn't a point source of light, this formula may underestimate the light at very low angles, and it doesn't account for either obstruction of part of the planet when it intersects the horizon or part of the planet still appearing above the horizon even when the center is below it.
- Do measures from an observer on the moon's surface to the planet's center, which is not quite the same as the orbital distance or semimajor axis for the moon, which is measured from the moon's center. For most cases where the moon's radius is small relative to Do, this makes little difference, so may not be worth worrying about.
- This formula assumes that the observer is far enough away that about half the planet's surface is visible and that all parts of the planet's surface are at about the same distance from the observer, proportionally speaking. This again should usually be true where the planet's radius is small relative to Do; for very close-orbiting moons, this formula will be progressively less accurate, but could still serve to give us a rough idea of lighting conditions.
One consequence of this approach is that the formula predicts some amount of reflected light at all phase angles except exactly 180°. In reality for the period of the total eclipse, we should expect the entire lit side of the planet to be hidden from the moon, though perhaps there could be some meager amount of light refracting through the edges of the planet's hemisphere.
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| A diagram demonstrating some of the important parameters of distance and angle, with dBS here corresponding to the formula's Ds and dBO to Do. Renerpho, Wikimedia |
The result of this calculation is illuminance of the moon's surface in lux, a generally convenient measure of visible light conditions. As points of reference:
- 100,000 lx: midday at the equator with clear skies
- ~10,000 lx: overcast day
- 100-1,000 lx: sunset, indoor lighting at night
- 1-10 lx: twilight, street lighting at night.
- 0.05-0.3 lx: clear night with a full moon
- 0.0001 lx: moonless, overcast night with no nearby light sources
What you might quickly notice is that conditions that appear "well-lit" to our eyes can cover a vast range of actual illuminance; our eyes are actually quite good at adjusting to different lighting conditions, such that we're neither totally blind at night nor overwhelmed by day. This also means that our perception of light conditions doesn't correspond terribly well to the actual amount of light energy acting to warm the climate or power photosynthesis (I sometimes see writers trying to convey planets seeming noticeably brighter or dimmer based on orbital distance, but for planets in the habitable zone at least there probably wouldn't be much apparent difference, just as someone travelling between Brazil and Iceland probably isn't going to notice an obvious difference in the intensity of midday light independent of weather and climate conditions, outside perhaps the dimmest periods of winter). We're assuming all these moons are habitable from the outset, though, so I'm mostly concerned with the perceived light conditions here.
When it does come to climate and such, though, note for convenience that with typical sunlight, 93 lx of visible illuminance corresponds to about 1 W/m2 of light power. This could vary between stars, and with reflected light may also somewhat depend on the color of the planet—and that also means the planet could tint the light, though again this may not be as obvious as you might think because our eyes are also quite good at adjusting to different colors of light; take a picture of a white sheet of paper under sunlight and under candlelight or an old fluorescent bulb and note how different the pictures actually look when viewed together compared to how much notice you usually take of differing color from these lights.
Anyway, we'll get into specific numbers in a moment, but the upshot of all of this is that reflected light from a habitable-zone planet onto nearby moons can easily reach into the hundreds or thousands of lux. That is to say, you could comfortably read and work in the middle of the night on these moons with just this reflected light, no need for artificial lighting. That means that for the planet-facing hemispheres of these moons, there may be no truly dark time of day except for eclipses. I don't think I need to emphasize too much the scale of consequences this could have for diurnal cycles of life, work and sleep patterns, perceptions of time, scheduling, and the need for artificial lighting, and so on. Reflected light levels will vary through the night as the phase angle changes, with light diminishing with larger phase angles: for areas near the center longitude of the planet-facing hemisphere (latitude shouldn't matter much here), reflected light illuminance will peak at midnight, when the phase angle is 0°, while sunrise and sunset happen near phase angles of 90°, when illuminance should still be almost 1/3 that peak value, so there may be no particularly dark period between day and night. At areas closer to the east and west edge of the planet-facing hemisphere, there will be periods of night at higher phase angles that may then be darker—but given how little illuminance is required to match twilight conditions on Earth, nights may still not get truly dark by our standards.
Put it together, and we start to get a sense of the daily patterns of lighting near the middle of the planet-facing hemisphere of a low-inclination moon: bright mornings like we're used to from direct sunlight, and then a sudden stark fall into darkness in the midday as the apparently static planet in the sky eclipses the sun. This, the darkest time of day lit only by stars (or perhaps other moons), lasts for a few hours before the sun suddenly returns at close to full strength. Then a normal afternoon and evening follow, with the sun sinking lower. The planet above is at first basically invisible with its dark side facing the moon, but the lit face slowly rotates into view, following the progress of the sun, and reflects light onto the moon. The sun's light still probably dominates until close to sunset, at which time there may be a curious transitionary period where all objects have two shadows from the perpendicular light sources.When sunset finally comes, there is no rapid plunge into twilight and then night; reflected light from the planet keeps the surface illuminated, slowly growing towards midnight, with outdoor lighting perfectly adequate for continued activity. There is still probably a clear difference: more shaded areas may noticeably lack good lighting, though shadows may be softer-edged if the planet is significantly larger in the sky than the star. Shadows also won't move much at night, aside from a slight shift as different parts of the planet's visible face are illuminated; areas shaded from the planet will never receive direct light in this period. It may become cooler, and wet surfaces won't dry as quickly. The color of the sky may shift, though stars may never become visible.
After midnight, when the planet's lit side is fully visible, sans perhaps a short period with the moon's own shadow passing across it—a frequent reminder to those below of their scale in cosmic affairs—the lit side then starts to turn away from the moon. The dimming is unlikely to be terribly obvious, with another gradual transition as sunrise comes and the world then brightens more substantially into morning.
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| A typical progression of day and night on the planet-facing side of a tidal-locked moon, through morning, eclipse, evening, and a planet-lit night |
That's the broad pattern we can expect, anyway. But to pin down the details a bit more precisely, I think it would be helpful to work through a few illustrative examples, and trace out the exact lighting patterns resulting from their parameters. For all of these I'll use a few common parameters, as well as make a number of simplifying assumptions, a few I'm reiterating just to make them clear:
- All the planets will occupy an Earth-like orbit at 1 AU around a sun-like star, such that they receive the same light as Earth and the star has an apparent diameter of 0.53°. Variation in distance from the star due to their orbits will be negligible for all these moons.
- All moons will be on circular orbits with zero inclination between the orbit of the moon around the planet and that of the planet around the star, and all will be perfectly tidal-locked to the planet.
- We'll ignore any atmospheric effects (either by that of the moon or the planet) that may influence light levels.
- We'll use the formula for reflected light even though it doesn't fully account for variance in angle of light from different parts of the planet or the exact portion of the planet that should be visible to the moon from different distances, but to avoid confusing matters, I'll set reflected light to be zero during total eclipses.
- Calculating illumination from a partially obscured star is a very complex exercise I don't much want to get sidetracked into, so for eclipses I'll take the highly simplified expedient of treating the portion of light from the star relative to what would be received from an unobscured star as following a sinusoidal curve from full at the start of a partial eclipse to zero at the start of the total eclipse, and vice-versa when the eclipse ends. For sunrise and sunset, I won't worry about the obscuring of the star by the horizon, we'll just treat it as a point source of light, and I also won't look too deep into cases where the planet is partially obscured by the horizon.
Julia
We'll start with our first example moon, Julia. Julia orbits a planet that is a direct clone of Jupiter in terms of mass (317.8 Earth masses) and radius (10.97 Earth radii), but we'll say it has a somewhat higher geometric albedo of 0.6. We will place Julia at a semimajor axis of 722,000 km (0.00483 AU), just outside the real orbit of Europa, to give it an orbital period of 4 Earth days. So in essence it's a habitable moon about as close as possible to the known patterns for formation of large moons, given our limited sample size. From Julia's surface, the planet has an apparent diameter of around 11°, about the apparent size of your fist if you stretch your hand out in from of you.
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| Relative apparent sizes of Julia's star (equivalent to our sun seen from Earth) and planet, as seen from Julia. |
We'll start at the sub-planetary point (SPP), the point in the middle of Julia's planet-facing hemisphere, directly facing the planet. As with all the cases we look at, we'll trace the illuminance across one full day, starting at sunrise:
What becomes immediately obvious is that A, in absolute terms direct solar lighting completely dominates over reflected light, and B, the midday eclipse is a significant interruption to this light, but a brief one, and doesn't dramatically reduce the total daylight the surface receives. Specifically, out of 48 hours of daylight, the total eclipse lasts about 2 hours and 50 minutes, while at least partial eclipse lasts a bit over 3 hours, 10 minutes.
To get a better sense of how these conditions might appear to humans (or other observers with similarly dynamic light perception), it may be better to look at illuminance on a logarithmic scale:
Peak illuminance just before and after the eclipse is over 126,000 lx, with most of the day over 10,000 lx, just as on Earth; the dimmest period outside of eclipses comes just after sunset and before sunrise, with about 140 lx from reflected light; still reasonably bright in comparison to indoor lighting, though there may be a brief period of adjustment from the more intense sunlight just minutes before. The transition here is fairly stark; only for the last 8 minutes of sunset is direct solar light less than 10 times greater than reflected light. Through the night, Illuminance from reflected light slowly climbs to a peak of 440 lx at midnight, but the difference is unlikely to be terribly obvious.
So, much as we said before: out of a 96-hour day/night cycle, only the ~3-hour eclipse is truly dark, with long periods of daylight beside it and perfectly sufficient lighting for continued activity throughout the night.
Much the same will be true for much of the planet-facing hemisphere:
At 45° longitude east of the SPP along the equator, the eclipse comes in the midafternoon, and reflected light from the planet is somewhat reduced. Just after sunset, only a crescent of the planet's visible face is illuminated, providing 15 lx of illuminance; just about enough to still see clearly and remain active, but a rather more apparent dip from the preceding direct sunlight requiring a bit more adjustment time. Improved lighting throughout the night will probably also be more noticeable, rising to a peak of 310 lx late in the night. Morning is more gentle, with reflected light still providing over 230 lx at sunrise.
At 80° longitude east of the SPP, near the edge of the planet-facing hemisphere where the planet appears to hang just above the horizon, there's a more substantial dark period of night. The eclipse comes in the late evening, after which there's only a bit over an hour of day remaining before sunset. When sunset comes, only a small sliver of the planet's lit side is visible, providing just 0.05 lx of illuminance, on the low end for a moonlit night on Earth. There's something like a full night for Earth here, with illuminance gradually climbing back up to 1 lx at 5 hours have sunset and 10 lx at 14 and a half hours after sunset. But with Julia's slow rotation, that still leaves 33 hours of night with dim but visible conditions, reaching 30 lx at midnight and peaking at 75 lx shortly before sunrise. One could definitely imagine a more substantial reduction in activity at night here, though perhaps resuming a bit more towards morning.
Note also that, with the planet so low in the sky, local topography and shading could substantially influence local lighting conditions; anywhere in this area with obstructions over 16° above the horizon in the planet's direction will never receive direct light from the planet at all.
Even farther east, past about 84° longitude from the SPP, are regions where the planet's disk touches the horizon, such that its light is partially obscured—with the most obscuration at the start of night, further deepening the gloom at those points. Beyond 96° longitude from the SPP, the planet disappears below the horizon entirely and nights are about as dark as Earth's, though of course really some areas might have slightly better or worse views of the planet based on topography, and a bit of light might refract through the atmosphere, so there could be a ring of the planet experiencing twilight nights even without direct view of the planet.
Towards the western edge of the planet-facing hemisphere, these patterns will all play out in reverse, with lighter periods after sunset, darker before sunrise, and eclipses in the morning. To the north and south, the daily patterns will remain the same along lines of longitude, but with progressively reduced light from both the star and planet; at 80° latitude north of the SPP, the profile of illumination will be the same as for the SPP, but with peak illuminance from sunlight just before and after the eclipse of under 22,000 lx, peak illuminance from reflected light at midnight of 76 lx, and minimum reflected light just after sunset and before sunrise of 24 lx.
Will any of this affect Julia's climate? Technically yes, but not by much. The eclipse lasts for about 3.1% of Julia's total day, so will reduce the total solar energy input relative to what it would get as a planet in the same orbit by the same amount, with reflected light from the planet adding back about 0.14 percentage points. 3% less light certainly isn't nothing; if Earth suddenly received 3% less sunlight, it would probably drop global average temperatures by at least 5 °C, perhaps substantially more after accounting for albedo feedbacks. But for comparison, a planet at the outer limit of the conservative habitable zone for sunlike stars would receive 64% less sunlight than Earth. So in practice we would expect this to be easily compensated for by the carbonate-silicate cycle adjusting CO2 levels. In Julia's case, the low axial tilt complicates matters, but if we just assume the same Bond albedo and atmosphere as Earth for simplicity's sake, about 1400 ppm of CO2 should give it the same average temperature.
Locally, the effect may be a tad more significant. The SPP loses about 9.8% of its potential sunlight to the midday eclipse and gains 0.4 percentage points back from reflected light, for a shortfall of 9.4%. That's about equivalent to the difference in average heating between Earth's equator and 26° latitude, (though this is far from a perfect comparison, both because Julia lacks season and because this is limited to the single point of the SPP rather than a full line of latitude). At 45° longitude from the SPP, the shortfall is about 6.6%. So again, not anything terribly dramatic or a real concern for the moon's habitability, but there could perhaps be some subtle influence on global temperature and circulation patterns.
The reflected light also isn't likely to impact vegetation much, given as it provides under 5 W/m2 of power at most. There is some photosynthetic life that can live on such meager light sources, but in this case it seems like it'd just make more sense to optimize for the far more abundant energy available by day. There might be some edge cases, though; in deserts, for example, photosynthesis is difficult because plants can't inspire the required CO2 without simultaneously losing water (whether this is a specific quirk of Earth plants or a necessary consequence of the low-energy lifestyles of any photosynthetic flora is a separate topic I may want to come back to another time). Some desert plants thus use workarounds like CAM photosynthesis, whereby they absorb all the CO2 they need at night, when the lack of direct sunlight and cooler temperatures far reduces evapotranspiration, and then store it for use by day. On the planet-facing side of Julia, it could perhaps make some sense for desert flora to try to eke out a bit more photosynthesis at night just because it comes at a much lower water transpiration cost. Velos
For the next two moons, I decided to try to optimize a little for maximum eclipses, which can generally be done by lowering the bulk density of the planet, because this causes moons to move slower in their orbits, thus lengthening the time to pass through the planet's shadow; and reduces the Roche radius, allowing moons to orbit closer without tearing themselves apart.
To find an appropriate planet, I used the planetsynth tool I've discussed previously, designed to model the sizes of gas giant planets; I set a low but probably still reasonable bulk metallicity of 0.1, atmospheric metallicity of 0.01, stellar flux equivalent to Earth, and age of 4.5 billion years, then tweaked the total mass to see how low I could get the density. Usually more simplistic estimates of gas giant size predict a minimum density somewhere close to Saturn's mass, but for a planet in an Earthlike orbit it seems that smaller is better, as the combination of low gravity and extra heat from the sun allows for the atmosphere to be inflated to very low densities. planetsynth only works down to a minimum mass of 1/10 that of Jupiter, so I settled for a planet of 32 Earth masses and 8.84 Earth radii, for a bulk density of just 0.26 g/cm3, even though some observations have suggested that there may be "super-puffs" of even lower mass and densities below 0.1 g/cm3. We'll once again assume a geometric albedo of 0.6.
If we want to maximize time spent in eclipse for a moon, we should place it as close as possible to place the largest portion of its orbital path in the planet's shadow. The predicted Roche limit for a moon of Earthlike density is actually less than the planet's radius, so our innermost limit is likely imposed by the potential for significant drag on the moon's orbital motions from the outer layers of the planet's atmosphere. This might not be much of an issue if planet and moon are mutually tidally locked, but to give at least a bit of clearance I'll place the moon Velos at an orbit of 85,300 km (0.00057 AU); that gives it a 12-hour day, and places just under 23,000 km between the closest point of the moon and the edge of the planet.
At this distance, the planet is monstrous in Velos's sky, with an apparent diameter over 80°—ranging from about 82° near the edge of the planet-facing hemisphere to 91° at the SPP, so it occupies over half the distance from one horizon to the other.
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| Relative apparent sizes of planet and star as seen from Velos, though it's hard to express on a flat screen the appearance of an object occupying such a large portion of the sky. |
Curiously enough, the eclipse lasts about as long as Julia's, around 3 hours, but here that's a quarter of Velos's full day/night cycle, removing half the potential daylight hours.
At the SPP, this removes much of the middle of the day; there are two brief periods of about an hour and a half with direct sunlight flanking the midday eclipse. But the moon does make the most of its night, with illuminance ranging from 7,500 to over 23,000 lx. That's likely to be an overestimate because significantly less than half the planet's surface is visible, and part of this visible surface is shaded by Velos itself (from the SPP at midnight, Velos's own shadow on the planet has an apparent diameter of about 32°), though on the other hand the visible parts of the planet's surface are also closer than the formulas here properly account for. Still, it does seem likely that the reflected light could be fairly bright, comparable to an overcast day on Earth, though still well short of even those shortened windows of direct sunlight.
At 45° longitude east from the SPP, the planet's apparent diameter has shrunk to 88°, so it's bottom edge just clears the horizon while its top edge falls just short of the zenith of the sky. There's a somewhat more substantial day of 3 hours that is then cut short just after noon by the eclipse, and then the sun briefly reappears for about 2 minutes before setting behind the horizon. Reflected light at that point is predicted to be 770 lx by our formula, but is probably actually substantially less, because again less than half of the planet's surface is actually visible and the lit side should be just coming into view at the end of the eclipse, so there may actually be a period after sunset with no better than twilight illuminance. Still, Velos's fast orbit means this won't last long and most of the night is quite well-lit, surpassing 10,000 lx shortly before midnight.
Any farther from the SPP and the obscuration of the planet by the horizon complicates our equations, but areas as far as 128° longitude from the SPP may still be able to see some part of the planet peeking above the horizon, so only a portion of even Velos's outward hemisphere will be free from its influence.
The potential climate implications here are rather more substantial. Velos is losing 1/4 of its potential sunlight to eclipses, and at best reflected light is only returning around 7 percentage points. That still leaves it potentially habitable, but if we wanted to shift its orbit it would have a notably different habitable range to a free-orbiting planet; whereas the sun's conservative habitable zone is generally taken to stretch from about 0.95 to 1.67 AU, in Velos's case the range of equivalent total light would run from 0.82 to 1.51 AU. Of course the dynamics of habitability are a little more complex than that, informed in particular by the albedo and climate of the surface.
At Velos's SPP, over 70% of potential sunlight is blocked by the eclipse, though reflected light might give something like 20 percentage points back, such that overall it receives about half the light of points on the opposite side of the planet, unaffected by eclipses. This is about the same as the difference in average heating between Earth's equator and 66° latitude, so we could expect a pretty substantial difference in temperatures, and perhaps a significant asymmetry in ice cover around the planet, leaving an oval-shaped warm region wrapping around the outward hemisphere.
Though, for parts of Velos's planet-facing side that are warm enough for plant life, over 10,000 lx of reflected light (~100 W/m) is a rather more substantial potential energy source for photosynthesis, and if that light is tinted a substantially different color from direct sunlight, that might favor different photosynthetic pigments, so there's potentially some opportunity for niche partitioning of different floral species optimized for different light sources.
Lesur
Next, we'll look at another moon of the same planet, a sister to Velos. Here, I want to maximize not the total time spent in eclipse, but the length of each individual eclipse. For this, we want to place the moon farther away from the planet, such that it moves slower through its orbit, though this has to be balanced against both the smaller angular diameter of the planet at greater distances and the limits of orbital stability. In this case, the optimum balance seems to be to place the moon close to the outer limit for stable prograde orbits; a moon in a retrograde orbit could be stable farther out, but would have shorter solar days and so shorter eclipses—though these limits are based on the particulars of this planet's orbit of a sunlike star, so conceivably there may be ways to optimize further, but I think this case is reasonably representative.
In particular, I chose to place the moon Lesur at around 2,200,000 km (0.0147 AU), which gives it an orbital period of about 65 Earth days and actual solar days of 80 Earth days. Such long days may imply a fairly extreme climate, though actually Lesur isn't far from the distance where it might be able to avoid tidal-locking for billions of years, so if Lesur were substantially younger than modern Earth (something like 2 billion years old) or somehow more resistant to tidal spindown, or if you did play with the numbers some more to get it to orbit a bit farther out, it might be reasonable for it have more rapid rotation (in which case the month length will be the same in terms of the apparent time to circle the planet relative to the sun, but the length of eclipse at any one spot might be slightly different, perhaps a bit longer, as the moon's rotation somewhat counteracts the motion of the planet's shadow across Lesur's surface; though if the rotation is fast enough, some areas may actually pass in and out of the shadow at points of the eclipse).
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| Apparent sizes of the star, planet, and Velos from Lesur, with Velos copying Earth's appearance and shown here at its maximum apparent distance from the planet. |
At this distance, the planet has an apparent diameter of just 2.9°—and while we're at it, Velos appears 0.33° wide and is separated from the planet by at most 0.59°. Small though this is compared to the experience of Julia and Velos, the slow motion of Lesur through its orbit means that when the eclipse comes, once in the 80-day orbit, the partial eclipse lasts for 18 1/2 hours and the total eclipse for 12 hours 49 minutes. Thus, the SPP experiences something like a full Earth night in the middle of its long day—though, because the whole month is so long, this is only interrupting less than 1% of the moon's energy input, so we really don't have to worry about climate implications here.
By night, the planet still manages to give significant illumination even at this distance, providing just under 10 lx immediately after sunset and 30 lx at midnight, placing the SPP a bit above twilight lighting conditions (Velos maybe adds another 0.1-0.25 lx depending on its position). This of course declines with longitude from the SPP, down to a range of 1-20 lx at 45° and 0.003-5 lx at 80°. Though, if the moon isn't tidal-locked, longitude shouldn't matter, with all longitudes reaching close to that full night illuminance at the right point of the month, but also having nights with no reflected light for close to half of the month when the planet passes outside the view of Lesur's night side—though illuminance at night would still decline towards higher latitudes, depending a bit on if the moon has any obliquity.
This all really requires the moon to orbit with very low inclination relative to the planet's orbit, though; even a mismatch of 2.5° is enough for it to sometime miss a total eclipse. At more substantial inclination, eclipses could be quite rare, as the window for potential eclipses could be much shorter than the 80-day month. But that could be interesting in its own right; a world with a prominent body in the sky (or a few including other moons, perhaps a whole celestial orrery) that usually has little effect aside from providing a bit of nighttime illumination, but in rare cases causes the entire world to essentially skip an entire day (presuming Earthlike rotation here), presumably causing all manner of distress and upheaval on the surface if the inhabitants don't understand the cause.
Geminar
For our last example case, I wanted to look not at a moon orbiting a larger planet, but one of a binary pair of identical planets. Geminar and its twin orbit each other at a distance of 53,100 km (meaning about 40,400 km between their surfaces at the closest), giving them both the same solar day as Earth. Each twin thus appears 15.7° wide in the sky of the other, and they experience total eclipses of about an hour (with partial eclipse for only another 4 minutes). Looking from one planet to the other, features as small as 20 km across might be just visible to the naked eye. Given that they're earthlike, we'll presume that both have a geometric albedo of 0.43.
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| Appearance of the star and the twin planet from Geminar, with Earth's surface copied at the correct perspective for viewing from the SPP |
Each planet casts a shadow about as wide as both itself and its partner, which has a couple important implications for the lighting pattern. For one, when Geminar's twin experiences its eclipse, this will reduce the light that can then reflect back at Geminar, to the point of completely interrupting reflected light at the point of maximum eclipse. Then, during Geminar's eclipse by the twin, the eclipse ends on the western edge of the twin-facing hemisphere at about the same time as it begins on the eastern edge. This means that, for most of the length of the eclipse, some part of Geminar's surface is still brightly lit by the sun and reflecting light onto the dark side of the twin. Some of this light may then reflect again off the twin back onto Geminar.
To some extent these effects should apply to all of our previous cases of moons orbiting planets; there should be a dip in reflected light as the moon's shadow passes over the planet, and during the beginning and end of eclipses there may be a somewhat lighter period when some part of the moon's surface is still lit and this bounces onto the planet and bounces again onto the dark part of the moon. But in those cases these were likely small effects, because the moons were quite small relative to their planets; thus their shadows only blocked a small portion of reflected light, and they spent most of their eclipses with the entire moon in the planet's shadow.
In this case it is worth accounting for, but trying to calculate this all out exactly is a very complex task, so I'm going to take a few shortcuts: much as we handled let's approximate the portion of reflected light from one twin onto another during an eclipse as following a sinusoid curve from 1 at the point the body first crosses into its twin's shadow to 0 at the point of total eclipse, when no light is reflected. This is probably not a great approximation but gives us at least a qualitative sense of how this affects lighting on Geminar.
At the SPP, the first 5 1/2 hours after sunrise proceed much like on Earth's equator at the equinox, and then the hour-long eclipse begins before noon. But rather than plunging into total darkness at the start of the total eclipse, the doubly-reflected light from the sun to Geminar's still-lit side to the dark side to the SPP still provides around 1 lx of illuminance. This is still quite a drop from the 125,000 lx just before the eclipse, and the illuminance will then decline, with most of the eclipse in the range of lighting of a moonlit night on Earth. Only at the point of maximum eclipse will this illuminance disappear, though this probably won't amount to a terribly obvious difference from immediately before and after.
Afternoon is then much as on Earth again, but then at night there is more substantial, singly-reflected light, beginning at about 200 lx just after sunset and then climbing to a bit over 600 lx half an hour before midnight. But then, reflected light starts to decline as Geminar's shadow crosses over its twin. Illuminance still remains above twilight levels until a period of perhaps a few minutes around midnight, with a moment of total darkness at the point where the twin is entirely covered in Geminar's shadow. The shadow then begins to pass off the twin, and reflected light illuminance returns, reaching the same peak and declining towards evening.
Thus Geminar features two short interruptions to its otherwise well-lit conditions, each varying somewhat between total darkness and twilight. This all does depend somewhat on the planets being identical in size; if Geminar were a bit smaller than its partner, then there would be a slightly longer totally dark period of the eclipse, but there would be no totally dark period at midnight, as Geminar's shadow would never completely cover the partner; if Geminar instead was a bit larger, there would always be at least a bit of doubly-reflected light through the eclipse, but a longer dark period at midnight.
At 45° east of the SPP, the eclipse comes in the midafternoon as you expect, but by the time the total eclipse starts, most of the rest of Geminar is already shaded, so there's less doubly-reflected light at the start and the totally dark period comes earlier, with a somewhat longer twilight period towards the eclipse's end, though not significantly brighter than at the SPP. At night the illuminance is of course a bit dimmer (ranging from 20 to 390 lx), and the dark period comes later in the night as well, but still with the same profile as for the SPP, as it's essentially globally simultaneous.
At 80° east of the SPP, the pattern continues, with both dark periods occurring later in the diurnal cycle and a more skewed distribution of doubly-reflected light during the eclipse. While we're counting doubly-reflected light, there's also a brief period after sunset where the doubly-reflected light off of the twin planet's dark side is greater than the single-reflected light from the visible crescent of its lit side, keeping conditions then slightly brighter, above 0.3 lx.
West of the SPP, the profiles are the same in reverse, with eclipses starting with a bit more illuminance and then getting dimmer towards the end. At higher latitudes, the profiles are of course consistent at each longitude but with all light sources dimming, which has the perhaps interesting implication that at high latitudes north and south of the SPP there would be 4 dark periods spread throughout the day: Geminar's eclipse, just after sunset, the twin's eclipse, and just before sunrise.
Though if the planets are of exactly equal size and there is 0 inclination in their mutual orbit relative to their orbit of the sun, there may be a significant region near the poles (above something like 68°) that never experiences a total eclipse, because as discussed before their perspective is slightly different such that the twin planet appears slightly offset in their skies, to the point that it never fully eclipses the sun. If there's some slight inclination, the poles may pass through periods of the year with total eclipses and only partial (or completely missed) eclipses, while the middle latitudes are always eclipsed. This does also imply that there likely is at least some doubly-reflected light reaching Geminar from the twin's poles at even the darkest points of its eclipse, but I doubt it amounts to much given the high angles of reflection.
The potential climate effects are slightly greater than for Julia, with the eclipse interrupting about 4.2% of the potential light to Geminar as a whole—though this may be a bit less if accounting for the only partially eclipsed poles—and 13% to the SPP, but I won't reiterate what I already said before.
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| The sizes and orbital distances of the 3 systems described at a scale of 2,000 km/pixel |
These examples hopefully give you some sense of the range of possible results you could get with habitable moons, or binary planets—and, as somewhat implied by Geminar's case, even a planet with a fairly sizeable moon might experience something similar. Julia is again probably the closest to a "typical" case for a moon, and Velos and Lesur may not be the absolute extreme cases but are not far off them, but ultimately we don't really know anything for sure about the formation tendencies of exomoons. Regardless, that'll do for this little dive into the topic; see you next time.
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