Q on satellite orbits + orbital drift

[JamesGoh]

Im aware that due to the oblate nature of the Earth, the satellite's line of apside and line of nodes rotate according to the following formulas (where i = angle of inclination of orbit to Earth and K is mean motion per day)

Variance in right ascension of ascending node ($$\Omega$$) due to rotation of line of nodes

d$$\Omega$$/dt = -Kcos(i)

Variance in argument of perigree ( $$\omega$$ )due to rotation of line of apsides

d$$\omega$$/dt = K(2 - 2.5sin(i)*sin(i) )

New argument of perigee (taking into consideration rotation of line of apsides)

$$\omega$$ = $$\omega$$o + d$$\omega$$/dt(t-to)

where $$\omega$$o is argument of perigee at epoch, to is time at epoch

In a textbook example I am reading, the author is trying to visualise the drift of the orbits as a result of just the rotation of the line of apsides (in other words i=90 degrees and the orbit is polar).

He starts of by assuming the situation where the perigee is exactly over the ascending node (in other words $$\omega$$ = 0 degrees) and d$$\omega$$/dt = - K/2. One orbital period (Pa) later, he states that the perigee would appear south of the equator (in other words $$\omega$$ = -KPa/2.

Mathematically, he is correct, but I cannot seem to visualise how the perigee would appear south of the equator. Would anyone be able to help me out here ?

If I am not clear, could anyone let me know ?

 

[Achy47]

it is important to understand and visualize the concepts and formulas we use in our work. In this case, the rotation of the line of apsides and its effects on the satellite's orbit can be difficult to visualize, but I will try my best to explain it.

First, let's define some terms. The line of apsides refers to the line connecting the perigee (closest point to Earth) and apogee (farthest point from Earth) of the satellite's orbit. The line of nodes refers to the line where the satellite's orbital plane intersects with the Earth's equatorial plane. The inclination angle (i) is the angle between the orbital plane and the equatorial plane.

Now, let's look at the formula for the variance in argument of perigee (dω/dt) due to rotation of the line of apsides. This formula shows that the rate of change of the argument of perigee (ω) is directly proportional to the mean motion (K) and the sine of the inclination angle (i). This means that as the inclination angle increases, the rate of change of ω also increases.

In the textbook example you are reading, the author is assuming a polar orbit (i=90 degrees) and a starting argument of perigee (ωo) of 0 degrees. This means that the perigee is directly over the ascending node. Now, when we apply the formula for the new argument of perigee (ω) after one orbital period (Pa), we get ω = ωo + dω/dt(t-to) = 0 + (-K/2)(Pa-to). This means that after one orbital period, the new argument of perigee will be -KPa/2.

To visualize this, imagine the satellite orbiting the Earth in a polar orbit with the perigee directly over the ascending node. As the Earth rotates, the line of apsides also rotates, causing the perigee to shift slightly to the west. After one orbital period, the perigee will have shifted by -KPa/2, which means it will now be south of the equator.

I hope this helps you visualize the concept better. If you have any further questions or need clarification, please don't hesitate to ask. As scientists, it is important to understand and be able to visualize the concepts we work with, so keep asking questions and