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Do any orbits around Earth leave the spacecraft in permanent shadow? If not, what is the minimum amount of time that a spacecraft must be exposed to sunlight on average?

fred_dot_u
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Slarty
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2 Answers2

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There is no orbit around the Earth that remains in permanent shadow. Such an orbit would need to have a period of one year, and that orbit would be too large to fit within the Earth's Hill Sphere, which represents a broad upper bound on the range of stable orbits around Earth.

In theory, an object near Earth-Sun L2 (at roughly 1.5 million km from Earth) could permanently be in Earth's shadow, but but being at Earth-Sun L2 is definitely not being in Earth orbit, and as Max Q Lagrange answered to another question, is too far from Earth to block the Sun completely. (Thanks, Engineer Toast, Naktibalda)

In general, on average, the best you can do is having the satellite spend a little less than half its time in the Earth's shadow, accomplished by having the object orbit as close to the surface of the Earth as possible. For a periapsis at 400 km over an airless, spherical Earth, this means a bit under 40% of the time in shadow.

Low orbit over airless, spherical Earth with shadow.

To answer CharlesStaats's question in the comments; Having a very elliptical orbit with the apoapsis in the Earth's shadow and using perturbations to keep it there doesn't help. The fraction of time that the satellite spends in the umbra continually decreases as orbital semi-major axis increases.

100Mm semi-major axis over airless, spherical Earth

With a semimajor axis of 100,000 km, the orbit is only spending 9.4% of its time in the shadow.

While the amount of time spent in the shadow initially increases as the semimajor axis does, so does the total orbital period, and the latter increases faster. Eventually, the narrowing of the umbra also comes into play, and the time in the shadow starts decreasing as well.

Raise the apoapsis to about the distance of the Moon, and the Satellite only spends about 5.6% of its time in darkness.

Geogebra Graph I made for the images, with semi-major axis slider

notovny
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  • So such an orbit is possible for a limited period but is destabilised because of its size and the disturbing influence of bodies such as the Moon on such extreme orbits? – Slarty Oct 08 '21 at 16:39
  • I used calculator at http://calctool.org/CALC/phys/astronomy/earth_orbit and calculated that an orbital height of 2,152,030 km is required to have a period of one year. that's 5.6 times more than an average distance to the moon. – Naktibalda Oct 08 '21 at 16:56
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    the shadow of Earth extends only 1.4 million kilometers according to https://en.wikipedia.org/wiki/Umbra – Naktibalda Oct 08 '21 at 16:58
  • @Slarty The one-year-period orbit around Earth is basically not possible at all. You wind up well beyond L2's distance at that point and the gravitational influence is wholly dominated by the Sun. – notovny Oct 08 '21 at 17:10
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    At least for a given time of year, couldn't you have a highly elliptical orbit with the periapsis on the sun side and the apoapsis in shadow so that you spend more time there? – llama Oct 08 '21 at 17:55
  • @llama Yes, you could, but on average, over the course of the year, an orbit like that spends more time in sunlight than one in low Earth orbit. – notovny Oct 08 '21 at 18:08
  • A statite could possibly stay in shadow all year -- statites are not in orbit but constantly providing their own acceleration -- but statites are meant to use solar sails to accelerate, and that doesn't do much if you're in shadow. – Ross Presser Oct 08 '21 at 20:25
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    If L2 is at 1.5Mkm and the umbra is 1.4Mkm long, something at L2 would have an interesting view of the sun. – Engineer Toast Oct 08 '21 at 20:32
  • @EngineerToast You're right, of course. When I googled the distance of L2, I thought it said 1.2 million km, rather than 1.5 million km. Will adjust. – notovny Oct 08 '21 at 20:39
  • Could you use an eccentric sun-synchronous orbit to keep the apogee permanently in shadow, thereby remaining in shadow a little more than 50% of the time? – Charles Staats Oct 08 '21 at 22:41
  • @CharlesStaats I'd probably have to do some calculations to verify, but I suspect that the nature of being in an elliptical orbit that has to fit the Earth under perigee means that, even if the apogee is in constantly in shadow, no elliptical orbit keeps you in the umbra for a larger fraction of time than an Earth-hugging LEO orbit. – notovny Oct 08 '21 at 23:02
  • Interesting, IIRC orbital speed is at it's lowest at apogee and fastest at perigee, that said the shadow is widest nearest the Earth and tapers so notovny is probably right – Slarty Oct 09 '21 at 08:47
  • @Slarty While the speed is slowest at apogee,, the required widening of the ellipse to fit Earth under the periapsis when increasing semi-major axis puts a lot of the ellipse also at a significant distance, resulting in an ever-enlarging region where the satellite is moving slower and slower. Even if the umbra remained at constant diameter, the fraction of time that the satellite spent in it would decrease as semimajor axis increased. – notovny Oct 12 '21 at 19:34
  • @notovny yes you are quite right I can see that now thx – Slarty Oct 12 '21 at 23:02
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One highly reusable observation is the following: For any plane going through the centre of mass, a satellite must spend some part of its orbit on one side of it if it spend some time on the other side.

And unfortunately, we can draw a plane where all the shadow is on one side:

shadow earth

The only loophole would be that the shadow revolves around over the course of a year, so if the satellite had an orbital period of a year, it could follow the shadow. Sadly, these fictional "one year orbits" don't exist, as they are so far away form the Earth that the gravitational influence of the Sun dominates completely. Co-rotating points balancing the gravity of the Earth and the Sun is a famous mathematical result, but the shadow of the Earth does not extend so far.

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