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If I have the position and velocity vectors of a satellite in an elliptical orbit for one point in time, then I can know its position in its orbit at any other time, and with that I can calculate the minimum altitude from the center of earth and the time it occurs.

Can we say here that we will get many similar minimum altitudes? Which means many periapsis points? In other words, can we say that the we have many periapsis points each point appears during a one complete rotation of the satellite around earth?

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    $\begingroup$ A Keplerian orbit has a single periapsis which is passed through each orbit, with more realistic modelling the periapsis will be different each orbit $\endgroup$
    – user20636
    Feb 29, 2020 at 23:46

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Can we say here that we will get many similar minimum altitudes? Which means many periapsis points? In other words, can we say that the we have many periapsis points each point appears during a one complete rotation of the satellite around earth? (emphasis added)

Yes, yes, and yes!

tl;dr: quoting @Polygnome's comment:

In other words, there is one periapsis per revolution, and for Keplerian orbits all revolutions are exactly the same, but not in the real world.

We approximate orbits around a body like Earth as being periodic and elliptical. that's not exactly true but I'll discuss that further.

First let's assume that the Earth were spherically symmetric (and not oblate and slightly lumpy) and ignore the effects of gravity from the Sun and Moon and other planets and atmospheric drag.

In that case orbits will be exactly periodic and closed, meaning for a period $T_0$ if the object is at some location vector $\mathbf{x}$ with velocity vector $\mathbf{v}$ at time $t$, then it will also be at those vectors at any time $t + NT_0$ where $N$ is any integer (e.g. ...-3, -2, -1, 0, 1, 2...).

We use the word periapsis in several different ways. We can use it to represent the "periapsis altitude" or the "periapsis position vector" only, or we could use it in to talk about whole set of events where it passes through that point.

Here are some examples:

  • "Periapsis is at 314 km."
  • "Periapsis altitude is 314 km."
  • "Periapsis will occur over the landing zone."
  • "Periapsis is the endpoint of the eccentricity vector."
  • "These days periapsis is usually on the night side of Earth."
  • "Periapsis always happens exactly one half period after apoapsis."
  • "Periapsis always happens exactly one half period before apoapsis."

However, in the real world, orbits around the Earth are not exactly closed (they don't return to exactly the same spot after one orbit) nor exactly periodic, because the Earth is oblate and lumpy, the Sun and Moon and other planets have gravitational effects, and there are weak effects like atmospheric drag and radiation pressure from sunlight.

So when very careful detailed calculations are done, while every orbit has a periapsis, it happens at a slightly different place and slightly different altitude, and the times between them is slightly non-periodic.

In this case we know the orbit isn't exactly an ellipse, but it's so close that we call it elliptical. We know the orbit isn't exactly periodic but it's so close that we can still talk about the period as if it were.

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  • $\begingroup$ OK, is this means that the value of position vector r is not accurate item for identifying the periapsis, because the Earth non spherically symmetric shape this r may be larger at the perigee than another point because it equals radius of earth + height of satellite?? $\endgroup$
    – Khaled Yassin
    Mar 1, 2020 at 4:42
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    $\begingroup$ @KhaledYassin no, I mean to say that thinking of the orbit as a perfect ellipse is not 100% accurate. Each time the objects makes an orbit there will still be a point that we call the periapsis, and it still defined as the point where the length of the position vector $\mathbf{r}$ is a minimum. But each time periapsis happens it will be in a slightly different position and distance than the previous one. $\endgroup$
    – uhoh
    Mar 1, 2020 at 4:53
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    $\begingroup$ Thank you very much for your deep explanation $\endgroup$
    – Khaled Yassin
    Mar 1, 2020 at 6:36
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    $\begingroup$ In other words, there is one periapsis per revolution, and for keplerian orbits all revolutions are exactly the same, but not in the real world. $\endgroup$
    – Polygnome
    Mar 2, 2020 at 20:05
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    $\begingroup$ You already did quite a good job explaining it ;) Come to think of it, the periapsis and apoapsis of an exactly circular orbit are undefined. But I suppose that is more of an academic problem then a real one. Thats a question I need to ask tomorrow, how flight software handles that. $\endgroup$
    – Polygnome
    Mar 2, 2020 at 22:18

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