orbit what is point is actually being orbited and with respect to what reference?

The point for gravitationally bound system of astro-bodies is their mutual center of center of mass: i.e., their mass-weight average position.

There is a formula for center of mass, but we will not write it down.

But for a sufficiently symmetric body it is the geometrical center.

When you hang/balance a resting object, it's the point directly above/below the pivot point.

But what is the rotation with respect to?

It is with respect to the observable universe (i.e., bulk mass of the observable universe). So there is a sort of absolute rotation.

This understanding is a modern one based general relativity, and so unkown to Newton and every one before the advent of general relativity in 1915. But not knowing general relativity did not prevent Newton everyone up to 1915 for doing celestial mechanics correctly with their then understanding.

What is an inertial frame is a reference frame in free fall in a sufficiently uniform external gravitational field unrotating with respect to general relativity.

All physical laws are specified relative to inertial frame, except general relativity which gave us our modern understanding.

When you want to apply a you reference it to sufficiently inertial frame.

Now the Earth is in free fall, and so reference frame attached to the Earth unrotating relative to the observable universe AND unaccelerated relative to the Earth is inertial frame.

But on the Earth's surface, we are rotating and yet we apply Newtonian physics everywhere on the Earth's surface.

In fact, there are non-inertial frame effects everywhere, but they too small to notice on the small scale without special observations, but they do affect weather: cyclones and anticyclones.

The correction for non-inertial frame effects are called inertial forces (e.g., the centrifugal force and the Coriolis force).

In dealing with gravitationally-bound systems of astro-bodies, people usually choose to the their mutual center of mass as the origin for the systems. I call these center-of-mass frames.

Newtonian physics dictates that the internal gravitation cannot affect the center of mass motion. But the internal gravitation certainly affects the motion of the astro-bodies.

If the EXTERNAL gravitation is sufficiently uniform it only affects the center of mass motion.

If it is not sufficiently uniform, then it does affect the internal motion by what are called tidal forces which on Earth cause the tides due to the EXTERNAL gravitation of the Moon and Sun.

The less the bound systems of astro-bodies are affected by tidal forces the more easily they form and persist, and we identify them as planet-moon systems, planetary system, multiple-star systems, star clusters, galaxies, and galaxies. They form a hierarchy of center-of-mass frames.

Below, we look the hierarchy.

Caption: A diagram of a gravitational 2-body system with the spherically-symmetric bodies orbiting in elliptical orbits the system center of mass marked by a red cross. The center of mass is, of course, at rest in the center-of-mass (CM) inertial frame defined by the gravitational 2-body system. The center of mass is also the common focus of the elliptical orbits. The other focuses for the elliptical orbits are just empty points in physical space with NO special significance.

Features:

1. The elliptical orbits of this gavitational two-body system are determined by Newtonian physics (what is universal about the physical system) and initial conditions (what is peculiar or individual about the physical system).

2. Gravity is, of course, the force that pulls the astro-bodies into orbits.

3. Now in astro jargon, an apsis (plural apsides) is an extreme point in the orbit of an astro-body.

   The two kinds of apsides are periapsis and apoapsis.

4. The periapsis (AKA pericenter) is the arrangement of closest separation and the term is also used for the closest separation distance.

5. The apoapsis (AKA apocenter) is the arrangement of farthest separation and the term is also used for the farthest separation distance.

   A physical fact for orbits is that astro-bodies move slowest at apoapsis and fastest at periapsis.

6. The apse line (AKA line of apsides) is drawn through the periapsis and apoapsis.

7. In the analysis of gavitational two-body systems, people usually use the relative orbit of the smaller astro-body to the larger astro-body.

   A relative orbit is also an elliptical orbit if the non-relative orbit is.

   You will have imagine the relative orbit since it is NOT shown in the diagram.

8. By normal convention, the relative orbit mean orbital radius (AKA relative semi-major axis) is defined to be

           r_mean = (1/2)( r_periapsis + r_apoapsis ), 

   where r_periapsis is the periapsis separation and r_apoapsis is the apoapsis separation.

9. The relative elliptical orbit with mean orbital radius r_mean has periapsis and apoapsis distances given by, respectively,

             r_periapsis = r_mean*(1 - e)   and   r_apoapsis = r_mean*(1 + e) , 

   where e is the eccentricity of the relative elliptical orbit.

10. The actual elliptical orbits relative to the center of mass are scaled down versions of the relative elliptical orbit. The scale radii for any epoch are

                 r_1 = r*(m_2/m) and r_2 = r*(m_1/m) , 

    where 1 is the index for astro-body 1, 2 is the index for astro-body 2, r is the relative separation distance, and m = m_1+m_2 is the total mass. As you can see, if m_1 >> m_2, we have r_1 ≅ 0 and r_2 ≅ r. This just shows that if m_1 >> m_2, we effectively have astro-body 2 orbiting astro-body 1 which is effectively at rest at the center of mass.

11. In most real orbits, astronomical perturbations cause noticeable deviations from exact gavitational two-body system behavior.

12. There are special-case names for the apsides of familiar astronomical objects (see Wikipedia: Apsis: Terminology). But yours truly thinks mostly these are overly fancy, except for the most familiar astronomical objects: e.g.,
    1. Earth: perigee and apogee, where suffix gee is derived from Greek Earth goddess Gaia.
    2. Sun: perihelion and aphelion, where suffix helion is derive from Greek Sun god Helios.
    3. star: periastron and apastron, where suffix astron is ancient Greek language word for star.

    Other than the example cases, if one wants a fancy name, yours truly suggests just you prefix the name by peri- or ap-: e.g., peri-Jupiter and ap-Jupiter.

Credit/Permission: © Tom Ruen (AKA User:Tomruen), 2016 / CC BY-SA 4.0.
Image link: Wikimedia Commons: File:Periapsis apoapsis.png.
Local file: local link: orbit_.html.
File: Orbit file: orbit_apsis.html.

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