# Can the Right Ascension and Argument of Perigee of a spacecraft's orbit keep varying by themselves with time?

I came across the orbital data for a low Earth orbit spacecraft and one thing which I am not able to understand is why does its Right Ascension of ascending node and Argument of Perigee keep changing every day.

Firstly, it seems extremely unrealistic to me, that they are being changed by manoeuvring the spacecraft because:

1. They are very costly affairs and for doing them over the period of months, one would need a mammoth amount of propellant.
2. I can't see what purpose it would serve.

So, to me it looks like they are either changing by themselves or the orbital data itself is wrong. I suspect the data being wrong as I haven't come across any phenomenon where orbital plane keeps changing and and also rotating by itself with time. It would require strong external disturbances for that to happen by itself and I doubt there are such strong and sustaining disturbances acting on the spacecraft. So, is the data incorrect?

Edit: I checked data for two more spacecrafts on the same website and they also seem to be behaving similarly. This is either something I don't know of, or the data is garbage.

## 2 Answers

You are correct to a point that the RA of the ascending node and argument of perigee won't change over time without some external force acting upon the satellite. In a simplified gravitational field, an object's orbital plane remains fixed.

Unfortunately, reality is a lot more messy.

Earth's gravitational field differs significantly from that created by a hypothetical point mass - this is known as the Geopotential Model. In particular, the Earth's oblateness (equatorial bulge), defined by the J2 coefficient, has a significant effect on orbits. It causes them to precess over time, changing the RA/longitude of their ascending nodes.

Furthermore, perturbing influences from other bodies in the solar system need to be accounted for when accurately plotting orbits - they can have small but non-negligible effects.

As to your second point, these effects can be very useful for satellites. For example, Sun Synchronous Orbits are precisely designed to precess at a rate of 360° per year, maintaining a constant angle between the orbital plane and the sun. This is ideal eg. for keeping a fixed illumination for surface observation.

See this related question on J2 pertubations for some more detail on the maths.

• I see. Just to be sure, is the data realistic? Like does the day to day variation seem realistic? – ModCon Apr 12 '19 at 11:55
• The data looks realistic. The change in RAAN is only around half a degree per day, which is a realistic precession for this inclination. The fact that the argument of perigee changes so drastically I think is due to the low eccentricity. Basically if your orbit is as good as circular, small perturbations can change the argument of perigee drastically. – Alexander Vandenberghe Apr 12 '19 at 13:23
• @Niket I agree with Alexander Vandenberghe's comment. The very low eccentricity is causing the rapid changes in the argument of periapsis – Jack Apr 12 '19 at 14:04

@Jack's answer is excellent, I'll just address @Niket's question in this comment a little further.

Just to be sure, is the data realistic? Like does the day to day variation seem realistic?

tl;dr: The drift makes sense and is within 1% of what we can easily calculate.

There are some of the orbital parameters plotted at the bottom of the page. The RA of the ascending node cycles through 360 degrees in about 60 days. That's about right for a ~45 degree inclined orbit in LEO, the ISS does that.

It's a big deal for the ISS because it means it cycles through periods where it goes into Earth's shadow every 90 minutes and then periods where it's in constant daylight as its orbital plane rotates around the Earth.

Let's see if we can calculate it. From this answer:

The first equation in Wikipedia's Nodal precession for the rate of precession $$\omega_p$$:

$$\omega_p = -\frac{3}{2} \frac{R_E^2}{(a(1-\epsilon^2))^2} J_2 \omega \cos(i)$$

depends on the parameters of the orbit ($$a, \epsilon, \omega$$, i) and the Earth's equatorial radius $$R_E$$ and its $$J_2$$ term.

Let's use 6378137 meters for $$R_E$$ (from this answer) and 1.0826E-03 for $$J_2$$ (from this answer).

The satellite's period $$T$$ in your data table is 15.59029 revolutions per day, or about 5542 seconds. Then use:

$$\omega = \frac{2 \pi}{T} = 0.0011338 \ \text{sec}^{-1}.$$

From this answer get

$$a^3 = \frac{GM}{\omega^2}$$

where GM is Earth's standard gravitational parameter of about 3.986E+14 m^3/s^2. That makes $$a=$$ 6768601 meters, or an altitude of about 390 km.

Plug those all in to the first equation, and we get $$\omega_p = -1.2149 \times 10^{-6} \ \text{sec}^{-1}$$ If we multiply that by 60 days or 5184000 seconds, we get -6.298 which is almost exactly $$-2 \pi$$ or one complete cycle, just what the plot shows!

The argument of perihelion at first looks like it drifts steadily then flips by 180 degrees around day 85, but that's actually a smooth shape change since the eccentricity hits zero and bounces back. That looks like a natural precession as well, and not an orbital maneuver.

So @Jack's answer nailed it.

Python script for the plot is available here: https://pastebin.com/rx1np9Mv • The calculation of precession rate was really helpful. I have two questions: 1. You said that it goes through periods where its in Earth's shadow every 90 minutes. Just to clarify, this will be true for half of the orbit right? Also, there's only one orbit where the satellite will be in constant daylight right? 2. I couldn't understand why the flipping in Arg. of Perigee is a smooth shape change. Let's take Jun 21 in data for example. It goes from 7 deg on Jun 21 to 276 deg on Jun 22. This looks like either a manoeuvre or erroneous data point. – ModCon Apr 12 '19 at 20:07
• @Niket thanks for your question! I'll address them a bit here but I think both of them are so interesting that you should consider asking them as new questions, just include a link back to this question for background. For 1) yes, sometimes they have an orbit that cycles through sunlight and darkness every ~93 minutes, seeing 15 sunrises and 15 sunsets per day, and sometimes they are in constant sunlight. 2) When the eccentricity drops to almost zero, the shape of the orbit is nearly a circle. When that happens Arg. of Perigee becomes noisy and almost meaningless. Ask new Q. for more detail – uhoh Apr 12 '19 at 23:26
• Sure, I will do that. This looks quite interesting. – ModCon Apr 13 '19 at 7:09
• @uhoh You're probably plotting the osculating elements, but if you calculate the actual LAN and argument of the perigee you'll see a very different shape for "Arg peri". Please, take a look here: cristianopi.altervista.org/as/ele_graphs-regre.html (select "ISS" and scroll down to see "Argument of perigee" graph). – Cristiano Apr 14 '19 at 16:41
• Oops! I wrote "ISS" because I see ISS in your answer. "Tiangong 2" should be fine for the TIANZHOU 1. – Cristiano Apr 14 '19 at 17:06