# Does the dawn-dusk orbit maintain the inclination with respect to the ecliptic plane?

I'm trying to understand the mechanics of sun-synchronous orbits, specifically the dawn-dusk orbit.

If I understand correctly, the RAAN rotates around the plane of reference of the Earth, which is the equatorial plane. For that to be true, the inclination of the orbit with respect to the ecliptic plane has to change? In such a way that at one end of the orbit, during peak summer in the northern hemisphere, the inclination of the orbit is counter to Earth's tilt (making the orbit pretty much 90 degrees wrt the ecliptic plane) while at the other end of the orbit, during the southern summer, the inclination adds to the tilt, making the inclination wrt the ecliptic plane a bit lower than 60 degrees?

This photo I found illustrates what I mean:

First, a word of caution: Take diagrams published in conference papers that have little or no explanatory text with a huge grain of salt. Admittedly, this diagram was from an invited paper presented at a very highly regarded conference, but still, I recommend that you take it with a huge grain of salt.

The solar beta angle as defined in the paper, the angle between the satellite's Earth-centered inertial angular momentum vector and the vector from the center of the Earth to the Sun, is a continuous function of time. Since it varies from 11° to -36° and then back to 11° over the course of a year for this particular orbit, the intermediate value theorem dictates that the solar beta angle must pass through zero degrees two times each year. Those two times each year are when the plane of the orbit is face on to the Sun.

As an aside, others use a complementary definition of the solar beta angle, the angle between the satellite's orbital plane and the line to the Sun. The two definitions sum to 90° (they are indeed complementary).

If I understand correctly, the RAAN rotates around the plane of reference of the Earth, which is the equatorial plane. For that to be true, the inclination of the orbit with respect to the ecliptic plane has to change?

That is correct. That the equatorial frame right ascension of ascending node rotates by 360° over the course of a year while the equatorial frame orbit inclination remains constant means that both the ecliptic frame right ascension of ascending node and the ecliptic frame orbital inclination vary over the course of a year.

• Thanks for the answer! To further understand how the orbit works: it seems like in one of the solstices, when the angle defined as beta is maximal (36 in the image), it might be the case that for a period of time a satellite in the dawn-dusk orbit has no view of the sun but rather is blocked by Earth. Am I right? I'm trying to reconcile all of this with a quote from the textbook "Orbital Mechanics for Engineering Students" (p.214 4th Edition): " Sun-synchronous orbits are those whose orbital plane makes a constant angle α with the radial from the sun" Commented Mar 1 at 19:01
• I mean in an orbit of ~600km for example Commented Mar 1 at 20:25
• @popeye Regarding "Sun-synchronous orbits are those whose orbital plane makes a constant angle α with the radial from the sun" -- that is incorrect, despite being repeated in many textbooks and all over the internet; it has even been repeated on this site. That can be modified to a nearly correct statement "Sun-synchronous orbits are those whose orbital plane makes a constant angle α with the radial from the fictitious mean sun". (continued) Commented Mar 2 at 9:05
• The true Sun can be above and below the equatorial plane by as much as 23.5°, and the projection of the Sun onto the celestial equator can be ahead or behind the fictitious mean Sun by up to 4°. Commented Mar 2 at 9:10
• Regarding your question about whether a satellite in a lower dawn-dusk orbit can occasionally be eclipsed by the Earth, that is correct. Dawn-dusk orbits are not particularly useful and are thusly rarely used. Commented Mar 2 at 9:12