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I know that the angular orbital velocity of a satellite in a circular orbit is $w_0=\frac{v}{r}$ but I want to know if the angular orbital velocity of a satellite in the circle orbit is constant. I mean that derivative from x from the angular orbital velocity of a satellite in circle orbit is it zero?

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    $\begingroup$ I guess the two objects are orbiting about a common center of gravity so you can't have both exactly true, but the difference will be negligible for a typical man-made satellite and a planet. True? $\endgroup$ Commented 2 days ago
  • $\begingroup$ @SpehroPefhany good job pointing that out. Negligible for typical (because gravimetry instruments do detect the variation but are not deployed on most satellites), man-made (because the effect is obvious only on geological time-frames) and a planet (because planets are by definition form a spherical homogenous body due to gravity). Not sure about polar orbit around a geoid though. $\endgroup$
    – Basilevs
    Commented 2 days ago
  • $\begingroup$ Are we assuming that the parent body is perfectly spherical; has uniform gravity, etc.? Are we ignoring the effect of other bodies, eg Sun, moon and other planets? $\endgroup$
    – MikeB
    Commented 2 days ago
  • $\begingroup$ @MikeB this is a Space Exploration site. Math questions are on topic, but serve as a tool, not a goal. Spherical planet in a perfect vacuum is for math.stackexchange.com $\endgroup$
    – Basilevs
    Commented 2 days ago
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    $\begingroup$ In the perfect, Keplerian-Newtonian spherically symmetric two-body simplification, both bodies have orbits of the same period and eccentricity about the barycenter, so angular velocity would be constant ( and if you're not in that situation, then the orbits will not be able to be truly circular) $\endgroup$
    – notovny
    Commented 2 days ago

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Since the orbital speed $v = \sqrt{\frac{\mu}{r}}$ of an object in a circular orbit around a body of constant standard gravitational parameter $\mu$ remains constant, the radial distance $r$ of the object in a circular orbit is also constant, and the angle between the directions of the radial distance vector $\vec{r}$ and the orbital velocity vector $\vec{v}$ is also constant for an object in a circular orbit $(\theta = 90°$) , the angular velocity $\omega_0 = \frac{v}{r}\sin \theta$ of the object around the central body is also constant.

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The speed of an object in a circular orbit remains constant but the direction of motion changes continuously.

As velocity is a vector composed of speed and direction, the velocity of an object in a circular orbit continually changes in line with the acceleration it experiences from the central object. It is continually accelerating in free fall, but by virtue of it's velocity is unable to reach the ground as it continually over shoots.

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  • $\begingroup$ The question was about angular velocity, though? $\endgroup$
    – Basilevs
    Commented 2 days ago
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Perfectly circular orbits are impossible.

Gravity

Irregular density and shape of a celestial body and its other massive satellites form a complex gravity field disturbing the orbit and affecting its speed. There are even techniques to evaluate mass distribution of a body by exactly measuring the speed of an orbiting satellite.

Density variation of a large body is usually small, so Earth orbit speed variation is measured in µm/s over 200 kilometers (could not find exact numbers).

Such a small effect does not affect space travel (although polar circular orbits do need regular adjustments), but disturbs passive orbits long-term further complicating solutions for three-body-problem.

Lower orbits are most affected by irregularities of the main body, while higher orbits are more affected by child bodies (moons, or other planets in case of a star orbit).

Drag and collisions

Atmospheric drag affects surprisingly high orbits slowing down (and therefore increasing angular velocity by lowering the orbit (!)) all orbiting bodies. Collisions with other satellites also affect both speed and direction of an orbit. This does affect space travel significantly, as constant boosts are required for any low orbit and collision avoidance measures are taken at high orbits.

Wind

Solar wind has major effect close to a star. Its charged component is highly irregular and relatively powerful, but photons also have some minor effect on planet's satellite (less so for star-centric orbits, where light pressure is more consistent). Oort cloud is probably affected by galaxy winds (caused by relative motion of the Sun through galaxy's medium, don't quote me on that, though).

Planets without magnetic field

The wind pressure is independent of orbit's radius but occlusion of a star and change of star-radial travel direction introduces an variability in orbital speed.

Magnetic field

Taking into account magnetic field is hard, but we can say that higher orbits have more wind pressure and it is more consistent, while lower orbits have less pressure on average with higher variability.

Winds do not affect space travel for now, but may be used with solar sails in the future (the perspectives are unclear).

Thrust

Artificial satellites have thrusters and may change the orbit height (and therefore speed) at will. Outgassing of natural objects may be ignored, as they do not do that for long to a significant degree in circular orbits (only important for highly elliptic ones like comets).

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Basilevs is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
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  • $\begingroup$ @LawnHollanderLawn I've incorporated some input from your edit, but disagree on important details like removed links and examples. $\endgroup$
    – Basilevs
    Commented 2 days ago
  • $\begingroup$ Yes, but this doesn't answer the question. The OP specifically asked for what happens in an idealized circular orbit. Those are indeed impossible in reality, but that doesn't mean it's not useful to understand the simple model first. $\endgroup$
    – Ryan C
    Commented 2 days ago
  • $\begingroup$ @RyanCthis implication is not clear to me. $\endgroup$
    – Basilevs
    Commented 2 days ago

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