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My question may sound naive and hence would like to apologise if this question is already answered in same or other form in this forum.
How does Right Ascension Of Ascending Node ( RAAN ) of a sun synchronous orbit earth orbit is related to its local time ? Suppose a spacecraft needs to be injected into a sun synchronous orbit of certain a,e,i and local time t combination from a particular launch site. Now for the same combination of a,e,i and local time t, if the satellite is injected on any other day from the same launch site, the RAAN would differ. Still the desired local time is achieved. As long as rate of change of nodal regression is maintained, does it matter whether the injection RAAN should have any particular value ?

Soumajit
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  • As "local time" you understand solar time on Earth at point directly below satellite's nadir, right? (because "local time" has also other meanings; in the fundamental meaning it's set through political/regulatory process and only very loosely related to daytime as result of orbital mechanics.) – SF. Jun 28 '17 at 23:11
  • @SF by "local time", I meant the equator crossing time of a spacecraft, – Soumajit Jun 29 '17 at 02:41
  • Yes, that's a point in time. But given in time of control center timezone (= on-board clock of the probe)? GMT? Political local time of the country that point of the equator belongs to? – SF. Jun 29 '17 at 08:09
  • @SF I think what I meant by local time matches more with your last option. To make it more clear 9:30 am local time in the morning is different from 12pm local time in the afternoon. – Soumajit Jun 29 '17 at 11:44
  • I would understand equator crossing time, LTAN Local Time of Ascending Node, as being the local solar time, not the political local time. – Puffin Jun 29 '17 at 11:53
  • @Puffin thanks. I was not able to frame it into appropriate technical words. – Soumajit Jun 29 '17 at 12:03

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The special property of sun-synchronous satellites is the precession, at pace equal to Earth's orbital period around the Sun (1 year). That means, the satellite's RAAN drifts by a full 360 degrees over the year, and as result, local solar time at the moment of crossing the equator remains constant.

if the satellite is injected on any other day from the same launch site, the RAAN would differ. Still the desired local time is achieved.

No - the RAAN would not differ. The two RAAN values from moments of respective injections would, but if you want the second satellite to achieve the same solar time as the first, your injection RAAN must be equal to RAAN of the first satellite at the moment of the second injection - significantly different from what it was at its own injection time, having drifted to the new value with precession since.

Pictured below are two sun-synchronous orbits (or thereabouts...) differing only by Right Ascension of Ascending Node values - by 90 degrees. This picture remains "true" - unchanged - regardless of time of day or time of year. There's the sunlit side and the night side, and one satellite passes the celestial equator at noon and midnight, the other - at sunrise and sunset (6AM, 6PM) - regardless of whichever country or ocean happens to be below at that moment, and regardless of date.

IF the date was 20th March, the day of Vernal Equinox, then RAAN of first satellite would be 0, second - 90 degrees. At any other day, the angle will differ - at autumnal equinox these values will be 180 and 270 degrees, respectively. That's because the sunlit side of Earth will be facing in exactly the opposite direction of the universe, being on the other side of the Sun. But the orbits will remain in their orientation relative to the direction of Sun and the terminator line.

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SF.
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    thanks for making it straight! So do you mean that local solar time is unique to a particular value of RAAN. In other words, 8:30 local solar time would necessarily have different RAAN than 9:30 local solar time ? – Soumajit Jun 29 '17 at 13:42
  • 8:00 1st January has different RAAN from 9:00 1st January (exactly 15 degrees away; 360/24=15 : 1 hour) . But 8:00 1st February RAAN is about 30 degrees from 8:00 1st Jan ( (360 ~= 365) /12 = 30 : 1 month). – SF. Jun 29 '17 at 14:06
  • Also note, this is only for solar local time at crossing the equator. Sun-synchronous orbits have varied (high) inclinations and the local solar time varies in other points of the orbit in a less straightforward manner. – SF. Jun 29 '17 at 14:08
  • Thanks once again! But still I am a bit confused ! ( pardon me if I sound very naive ) . So the RAAN on 1st January is differing from RAAN on 1st February at 8:00 for the same solar local time ? – Soumajit Jun 29 '17 at 14:25
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    @user19925: RAAN is defined in relation to "distant stars". Solar time is defined in relation to direction of Sun from given point on surface of Earth at given moment. In half a year, direction of Sun for the same hour shifts by 180 degrees in relation to distant stars (Earth on the opposite side of the Sun). For 12 hours, direction of the Sun shifts by 180 degrees in relation to a point on Earth equator. Compare synodic and sidereal day definitions. Orbital elements are defined in sidereal; solar time is synodic, the quirk of sun-synchronous is maintaining synodic, not sidereal orientation. – SF. Jun 29 '17 at 14:32
  • If we were to define the orbital elements of the satellite in synodic system of coordinates (making a full 1 rotation per year in relation to sidereal) then RAAN of sun-synchronous satellite would be unchanging, solar time would be unchanging, and launching another at a different time would require exactly the same RAAN - or, for different solar time, shift by appropriate angle, 360 degrees per 24h. But RAAN is defined in sidereal system of coordinates, so you must adjust for day of the year, the 360 degrees per 365 days. – SF. Jun 29 '17 at 14:41
  • Thanks once again for insight answer! Now I am able to separately appreciate RAAN and local solar time. But still having difficulty in understanding the relationship between them. I think I should first understand the Local time of ascending node. – Soumajit Jun 29 '17 at 15:53
  • @user19925 If I understand it correctly, it's solar time at the point on equator directly below the satellite, at the moment the satellite crosses the equator. It may fall on different places on Earth, but it will be always the same hour there, then. Say, on Monday the satellite crosses above Mount Kilimanjaro, around 8AM Tanzanian time. About 12 hours later it passes over Andes in Ecuador, at around 8AM Ecuadorian time. "around" because the solar time at Mt. Kilimanjaro is off by a little from official Tanzanian local time - it passes at 8AM solar time sharp, for the place it passes. – SF. Jun 29 '17 at 16:14
  • Thanks once again! But in the above scenario where does a particular value of RAAN fit ? – Soumajit Jun 29 '17 at 16:29
  • @user19925: As a number - nowhere really, just "angle between vernal equinox and the ascending node" - it's a way to describe the orbit in a way that's not bound to whatever happens on the surface, or where Earth happens to be in relation to the Sun. It's very useful in satellites where precession is very slow, or actively counteracted to be zero, providing one of a unified set of orbital elements to describe any elliptic orbit. – SF. Jun 29 '17 at 17:28
  • With sun-synchronous It follows the synodic:sidereal shift 1:1, with a constant offset which decides which local solar time hour the satellite crosses the equator. At the day of vernal equinox, with RAAN=0 the satellite will cross the equator at local noon and midnight. With RAAN=15 - at 1AM and 1PM. But a couple days later RAAN will be different (even though the satellite still crosses the equator at these "local" hours). Regardless, for sun-synchronous it's a "derived value" to be put in databases and simulations, not something you design the mission around. – SF. Jun 29 '17 at 17:37
  • (compare to e.g. Molinya orbit, where RAAN is pretty important, defining which area of Earth will be observed - but not at which hours.) – SF. Jun 29 '17 at 17:39
  • Thanks a lot once again! I think I got the difference somewhat better than when I posted the question :) – Soumajit Jun 29 '17 at 18:00
  • @Soumajit: One way I think may help you imagine it: first pick the set: Sun-Earth, immobile, in an empty universe (no other stars or planets) with Earth being a completely featureless globe (you can't tell where the continents are, if it spins, or whatever. (but one side is sunlit) Then add the satellite, in a constant, non-precessing almost polar orbit. Mark the point where it intersects the equator, and a point on the globe below. Angle to that point from "noon" (line between Earth and Sun) is your corresponds to local solar time. – SF. Jun 29 '17 at 23:09
  • If it's 90 degrees - on terminator line - it's 6 AM or 6PM. If it's 15 degrees, then it's 1 hour off. And so on. This is your local time at ascending node, and this is a constant - that doesn't change or depend on anything else. Now if you add continents and Earth is spinning, the point will be over different points on Earth, but it's still unchanged. It's what happens to be below it when the satellite passes through it that changes. – SF. Jun 29 '17 at 23:12
  • And now, instead of making Earth spin around the Sun, add the whole universe spinning around the Sun-Earth system. We can, it's just a different frame of reference. Nothing changed. Pick one arbitrary point in the universe, oh, one star, and call direction from Earth there "vernal equinox". This point spins around Earth, and RAAN is the angle between the line [center of Earth - Ascending Node] and the line [center of Earth - Vernal Equinox]. Again, nothing changes, you just got one extra number, an angle to arbitrary point. – SF. Jun 29 '17 at 23:16
  • thanks a lot once again! Can you tell me few examples when a particular value of RAAN does indeed become important during injection and there after ? – Soumajit Jun 30 '17 at 02:34
  • @Soumajit: Say, Molinya constellation. 3 satellites differing only in RAAN and true anomaly; strongly inclined, strongly eccentric orbits - a bit like three petals of a tulip flower. These need to be three launches, so three different launch times, and two consecutive insertions need to be for RAAN +120 and -120 degrees from the first. (the difference between true anomalies is a little more tricky, 1/3 orbital period apart time-wise) – SF. Jun 30 '17 at 07:29
  • @SF thanks once again ! Then for the above 3 satellites a , e and i are same and only RAAN is different ? – Soumajit Jun 30 '17 at 12:41
  • @Soumajit: approximately :) I'm not sure if the constellation as a whole is not inclined to match Earth axial tilt, never dug deep enough to find out. If it isn't, then concerning shape of orbits, only RAAN is different. True anomaly (position in orbit at given time) would differ so that each satellite is in optimal position of the orbit when the respective "interesting" part of Earth is visible - Molinya was designed specifically to continuously observe 1. Siberia, 2. Canada and northern US. so each satellite is in apogee while above one of these territories. – SF. Jun 30 '17 at 12:56
  • @SF thanks once again ! Hypothetically if I assume 3 sun synchronous satellites to have same a , e and i and differ only in RAAN, then how exactly they are different from each other. – Soumajit Jun 30 '17 at 13:11
  • @Soumajit: See the edit to the question. – SF. Jun 30 '17 at 14:26
  • @SF thanks a lot one again! From your diagrams it looks like ( 1 ) If 2 sun synchronous orbits differ in the their local solar time, then they would necessarily differ proportionately in terms of their respective RAANs, ( 2 ) The injection RAAN would continue to drift maintaining the local solar time. – Soumajit Jun 30 '17 at 16:49
  • @Soumajit: Correct. And while RAAN drifts over the year, difference in two RAANs between two sun-synchronous satellites doesn't. – SF. Jun 30 '17 at 16:58
  • @SF thanks once again! They have helped a lot in gaining insight into RAAN. I'll try to converge towards my original doubt :) Now hypothetically suppose today I have launched a spacecraft into a sun synchronous orbit with 9:30 local solar time with particular value of RAAN r. Tomorrow the RAAN of the present satellite would be r+1 maintaining the local time . Now I I try to launch a new sun synchronous satellite at the same injection time as that of today ( targeting the same local time ) , how would the achieved orbit of the new spacecraft be different from the present one. – Soumajit Jun 30 '17 at 17:37
  • @Soumajit: It wouldn't - because our clock is based on synodic, not sidereal period. Between yesterday's noon and today's noon, Earth rotated about 361 degrees, not 360, so if you use timing, not coordinates to guide the launch and insertion, you insert the satellite into the same orbit (RAAN difference=0). – SF. Jun 30 '17 at 17:53
  • (the above only true for sun-synchronous obviously). – SF. Jun 30 '17 at 18:01
  • @SF thanks once again! So then between 9:30 local solar time today to 9:30 local solar time tomorrow, and between 9:30 sidereal time today and 9:30 sidereal time tomorrow, what is the difference in the shift of RAAN ? – Soumajit Jul 01 '17 at 03:44
  • @Soumajit: actual year is 365.2422 days which is adjusted through leap years to 365/366, so it's not that exact if you're using earth-based clocks (1 year doesn't add up to 24*365 hours), but $360 \over 365.24$ degree for synodic, $360 \over 366.24$ for sidereal. – SF. Jul 01 '17 at 14:24
  • Let us continue this discussion in chat. – Soumajit Jul 01 '17 at 14:51