For a SciFi project I work on I have a space-station orbiting earth that uses an 8 hour 3 shift system for its staff. Each shift is assigned to one-third of the countries and organizations on earth - namely those that are facing the station during the ~8 hours of the shift.
The above premise requires the station to be placed so that it will stay relatively put in regard to the earth and one of her full rotations.
Q: Where can I put my station so it can observe a full rotation of earth?
From what I understand your requirements are thus:
In astronomy and orbital mechanics, Lagrange points are positions in an orbital configuration where an object remains fixed relative to the other bodies. Point 3 automatically removes typical orbital positions such as Low-Earth-Orbit and Geosynchronous-Earth-Orbits. This leaves the Lagrange points as a natural selection; but, there are 4 such points for Earth and The Moon and 4 more for Earth and The Sun.
If we look at the Earth-moon system, L1 is a natural point to choose for the station. It's situated between Earth and The Moon with relatively quick access to either body and can observe Earth's full rotation with ease. That being said, L1 poses 2 issues:
If we look at the Earth-Sun system, many of the same points for the Earth-Moon system still apply. The only major difference here is distance. For Earth-Moon, L1 is 326,390 km as measured from Earth to the Moon. For the Sun-Earth system, L1 is at 57,689,000 km as measured from Earth to the Sun. This would place the satellite well outside the Earth-Moon system.
For the Sun-Earth L1 point, the 1st argument applies equally; but, the 2nd argument is nearly eliminated. The orbital period of the Earth-Sun Lagrange points is 1 year or 365.25 days. Therefore, during a single 24 hour station day, the station and earth will have moved about 1 degree around the orbit. This comes out to about 1 day for Earth. Technically, after about half a year, at 00:00 hours the other side of earth will be visible; but, now this is minimized dramatically.
Unfortunately, L1 for the Sun-Earth system experiences a lot of electromagnetic radiation.
If the station were placed at L2, it'd keep a close synchronicity with Earth's rotation without experiencing the EM radiation at L1. I will note that station would be 176x farther from Earth than at the Earth-Moon L2; but, whether or not this is an issue for your world is up to you.
Technically, I assumed Earth's orbit was perfectly circular. In fact this is not true; it's slightly eccentric. As a result, when Earth is closer to the sun during the northern winter months, it travels much faster therefore passing through more radial degrees per day and when Earth is farther from the sun during northern summer months, it travels slower. As a result, my calculations won't be exact; however, given the astronomical scales involved, the error range is small enough to qualify as a back-of-the-envelope calculation.
Here's a diagram I drew up:
Fundamentally, because L1 orbits with the moon, for every 24 hour period, points A,B,C,D will always return to the same position; but, the station will orbit 13 degrees to the marked L1 positions, each one above points A, B, C, and D respectively. Now suppose we have three shifts, each representing countries in the Cyan (CY), Magenta (MG), and Yellow (YW) nations below; and, suppose that the shifts are marked as CY from 00:00-08:00, MG from 08:00-16:00, and YW from 16:00-24:00. This means that after 9 days, the Earth will have rotated 9 times, and be at the same position; but, the station will have traveled 119.7 degrees around earth, or ~ 1/3 of its orbit. As a result, when CY begins their shift at 00:00 hundred hours, they'll find that directly below them is the start of MG territory. Normally this would indicate that they are ending their shift and MG's shift is about to begin; but, the station time is 00:00. If you choose L1, your station will need to use a 23.11 H day with shifts every 7.7 Hours (7 hours 42 minutes) to ensure shifts start and terminate above their respective territories.
Given the requirement to view the entire Earth once in a 24 hours period I can see only two solutions.
You can either gain 15 degrees (1 hour) per hour on the rotation of the planet, or lose 15 degrees per hour.
The former requires you to effectively (and exactly) "lap" the planet in a single day of its rotation, meaning you need to move precisely twice as fast. This puts you into a 12 hour orbit around the planet, meaning you make 2 orbits for the Earth's single rotation allowing you to see each point on the Earth by seeing 12 hours (180 degrees) of it on each orbit. Orbital distance 20,200km (approx). This allows the "near Earth" requirement and is the stable, safe option.
The latter forces you into a solar orbit as you have no effective movement relative to the Earth1. You can't just share an orbit though, you need to sit in one of the gravity neutral locations or you'll end up hitting the planet you're sharing orbit with. These safe locations are known as Lagrange points and are numbered L1 through L5 based on location, seen in the image below outlined by gravitational contours.
L3 is on the wrong side of the Sun so we'll write that off. L4 and L5 are 60 degrees ahead and behind the planet respectively and they are stable but put you too far away to be useful, they also have a downside in that they're often occupied by trojans. L1, L2 are closer, but unstable, L2 being more stable than L1. This breaks the "near Earth" requirement, but is apparently what you're looking for.
As a side note, NASA advises L5 for long term habitation
1 This is a three body problem, the bodies being the Sun, the Earth, the space-station. The Sun is the datum used to define the relative movement of the other bodies, not the rotation of the Earth.
The title of the question, and the actual question, do not seem to match up.
If different countries are visible during different parts of the day, then the station is not remaining "put". If you want it to "stay put", you need a geostationary orbit, at an altitude of 35,786 km.
If, on the other hand, you don't want it to "stay put", but you want it to pass over the same part of the Earth at the same time every day, that's quite different. In particular, it looks like you want it to make one revolution with respect to the co-rotating frame of the ground once per day, so that it passes over each line of longitude once every 24 hours.
There are two ways to do that: you can approximate it by orbiting really far away- like, as far out as the moon, or farther, so that for all practical purposes the station is completely stationary, and the Earth turns beneath it. If you place it at any of the Earth-Sun Lagrange points, that will work out perfectly- but the closest of those (the L1 point, on a line between the Earth and the Sun) is about 1.5 million km away. Not terribly practical if you need to communicate with the ground in near real time, or if you need to see things clearly.
Probably the better solution is to use a 12-hour sun-synchronous, (aka semi-synchronous) orbit. This way, after one orbit to the same position with respect to the sun (about 2 minutes longer than one sidereal orbit), the Earth will have spun half-way around, and you will have passed over 1/2 of the Earth along the way. And the second orbit, you will pass over the other half of the Earth. Your three shifts will thus be split up over 2 orbits.
The approximate altitude for such an orbit is 20,200 km.
Okay, according to this site, one of the formulas for calculating properties of a satellite's orbit is the following:
T^2/R^3 = (4*pi^2)/G*Mcen
T = orbital period in seconds
R = avg Radius in meters
G = gravitational constant
Mcen = mass of the body to be orbited around (in this case Earth)
We know G and Mcen, and we want T to be such that from your point of view on Earth, it takes 24 hours to orbit. We could either orbit slower than Earth rotates, but we'd have to be so far away we're effectively stationary, which has been covered in other answers. The other option is to go faster than earth rotates (specifically twice as fast), thus we'd need to orbit twice in 24 hours which gives us an orbital period of 12 hours.
This is because orbit and space and what-not is all described from a point of view. It's why people thought the sun orbited the earth for a while, before we realised we orbited it and rotated to cause the same effect. From the Moon's point of view it stays still and the earth slowly spins in place in the sky. From Earth's point of view, your stations passes through the sky exactly once a day, at the same time. From some distant observer outside of the solar system the earth spins once every twenty four hours and your station orbits it once every 48. Movement in space is all about perspective.
Thus, inserting all the numbers: (12 hours equals)
43200^2/R^3 = (4*pi^2)/(6.673 x 10-11*5.972 × 10^24)
Multiply G and Mcen:
43200^2/R^3 = (4*pi^2)/3.9851156*10^14
Multiply each side by GMcen and R^3:
43200^2*3.9851156*10^14 = 4*pi^2 * R^3
Multiply out the left side
7.43718*10^23 = 4*pi^2 * R^3
Divide both sides by 4*pi^2
1.88386*10^22 = R^3
cube root everything
26608243.6243 = R
We now have our average orbital radius of 26,608,243.6243 meters. This is from the center of the Earth though. Earth's radius is 6.371 million meters, so subtract that and we get an approximate altitude for your station of 20.2 thousand km.
This is a kind of alternative to KareemElashmawy's answer.
As they point out, the best you can do with a single station is to put it in one of the Lagrange points, but these have problems.
The Earth/Sun L1 point is too far away from Earth, and has fairly high radiation.
The Earth/Sun L2 point has lower radiation, but is also far away, and you'll only ever see the night side of earth.
The Earth/Moon L1 point would be close enough, but you'd be tied to the Moon's day/night cycle, which is 28 days long, and so half the cycle you'd see the Earth's day side, and half the cycle you'd see the Earth's night side, which isn't ideal.
None of these points are stable, and so you'd have to constantly monitor your position and apply corrections to stay on station.
Unless you want to use magic level tech you can't have an orbit that stays solely between the Earth and Sun and is within the orbit of the moon. You would essentially have to constantly expend thrust in order to hover in that position.
So I'd like to propose a "PQ" solution; Have three space stations in geostationary orbit. Each station would stay active while the section of the globe that it's monitoring is in daylight. This has a side advantage that there would be no back side of the planet that one of the stations couldn't see which could be useful for emergency and traffic control purposes.
You want your station to observe one full rotation of the earth. I take that to mean that in one day, persons on the station see the entirety of earth pass beneath them.
is a circular orbit 35,786 kilometres (22,236 mi) above the Earth's equator and following the direction of the Earth's rotation.
It helps me to see it. Here is my depiction, with the Earth as a clock face. The station orbits keeping 12 beneath it as the earth turns. To stay above 12 the station must orbit earth in 1 day.
Now use that same 24 hour orbital path at 35786 km that the geostationary satellites use, but go retrograde. These are dangerous orbits!
Nevertheless, a satellite in retrograde orbit could pose a major hazard to other satellites, especially if it was placed in the Clarke belt, where geostationary satellites orbit.
Now the satellite moves contrary to the rotation of the earth. At 12 hours the satellite will be again above its starting point, having seen half the entirety of the earth rotate beneath it. Then it sees the other half entire earth move past again and at 24 hours is again above its starting point.
This answers the question in that the station does see one rotation each day. And then another one, the same day. Question does not specify "one and only one" rotation seen per day.
Ummm... okay so I think the only solution is to put it in a separate but Earth oriented orbit so either L4 or L5 in a Trojan orbit, or L3 on the far side of the Sun. All these positions allow for uninterpreted observation of the Earth and a stable spacial relationship between the station and the Earth but none of them orbit the Earth. Any orbit of the Earth is going to have an offset period so the station won't be able to operate on an Earth matched clock, the station will go around the Earth in 12 hours while the world turns or some other none observation matched period. L1 and L2 also fit the bill in terms of orbital stability but they are not themselves stable because of Lunar gravity, they are much closer to the Earth though and the cost of propellant to maintain position in the lagrange zone might be worth it to reduce signal delays (light speed signals need 32 minutes for a round trip at L3, something on the order of 11 minutes for the Trojans, 10 seconds for L1 and L2).
Actually a 48 hour equatorial orbit (along the true rotational equator) that chased Earth's orbit should have a "lap time" of 24 hours, I think that fulfills your needs. So orbital radius approximately 67000km, velocity 2.5km/s, aligned with Earth's rotation, someone will probably point out why this doesn't actually work but it works in my head and appears to work on paper. The station goes around the Earth in 48 hours but it's overhead times will match up on a 24 hour cycle.
You mentioned this is a Sci-Fi project, so I'm going to approach this question more as a writer than an astrophysicist. The simplest way I could explain the mechanics would be to initially place the station above the international dateline. Then maintain a geosynchronous orbit opposite to the rotation of the Earth, but only so much so that in 12 hours, the international dateline (because the line circles the globe) will pass again under the station. In 24 hours, the station should return to where it initially was above the international date line.
A slightly different answer than the others seem to be.
Two potential locations for a station to orbit the Earth and be a near match for your requirements.
One is to build the station on the Moon on the side perpetually facing Earth. The other is to set your space station in the same orbit as the Moon, but on the opposite side of the Earth. This will likely take some active station-keeping to maintain the orbit.
For both of these locations, I believe the time it will take for the station to again be directly over the same point on Earth is just under 25 hours (split into three 8 hour, 20 minute Shifts, which includes slight overlap).
This does mean that there will be a gradual shift of which Shift is responsible for which parts of the planet. However, if it is absolutely required that the same people cover the same sections of the planet, it would be fairly simple to set up shift schedules to move personnel from one Shift to another, to match up with their geography.
With redundant personnel in each Shift (required anyway for illness, time-off, vacations to Earth, and random alien abductions), it is absolutely trivial to schedule such a change.
It is not really clear what you exactly mean by "it will stay relatively put in regard to Earth" (I also failed to understand the linked reference).
Please clarify what alternative you need.
