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Please excuse me for a lay question.

As we know, to calculate the orbital velocity, we take the mass of the orbiting body, the mass of the body being orbited and the distance between the two bodies into account.

However, we don't really care about the natural satellite(s) (of a planet) in the calculations. Why are they insignificant? Why isn't the whole system involved in the calculations?

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Yes, it does affect the planet's orbital velocity. Both planet and satellite move around the planet-satellite center of mass. That center of mass is orbiting the star with some constant velocity (let's assume circular orbits to keep things simple) but if you're only looking at the planet, then the orbital speed around the star will wobble up and down around the because of its additional motion around the planet-satellite center of mass.

For most planet-satellite systems this effect is very small since usually the satellite is very, very light compared to the planet. Only when you have a genuine double system like Earth-Moon or Pluto-Charon does this effect start to matter. For transiting exoplanet systems they're trying to infer the existence of exomoons by exploiting the effect this will have on the transit timing of the planet. So far no exomoons have turned up though.

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One way to look at this question is to not think of the Moon orbiting the Earth but look at it as two bodies orbiting each other, then it becomes obvious that orbital velocities of two objects around each other does affect the orbital velocity around the star.

The Moon and Earth both orbit the barycenter between them, which is inside the earth, but in the direction of the moon.

The moon orbits the barycenter about 3,640 km/hr (slightly slower than it orbits the Earth I think), so the corresponding earth's movement for 81 times the mass is an ellipse 1/81st the size, giving the Earth an elliptical orbital velocity of about 45 km/hr, which, being in an elliptical motion is a vector addition of velocity and so it's only about plus or minus 45 km/hr at full or no moon, when the motion is relatively parallel with, or 180 degrees opposite to the Earth's motion around the Sun.

see picture:

enter image description here

This velocity variation has a period of about 27.3 days (sidereal not synodic) with an average diameter of 1/81st the Moon's orbital diameter, or about 9,400 KM, which is about the same distance the Earth orbits around the sun in just over 5 minutes, so, the effect is tiny, and is probably tiny for every planet-satellite system, but feel free to calculate Jupiter-Ganymede or Pluto-Charon if you like.

Using the 30,000 km/s approximation, we get 30,045 km/hr at no moon and 29,955 km/hr at full moon or about 3/10ths of 1%, peak to troth, every 13.65 days or so.

For comparison, The Earth's perihelion is about 3.28% closer to the sun than it's aphelion, which using Kepler's equal areas law and an approximation of area = 1/2 base x height, the Earth's orbital velocity is about 3.28% greater at perihelion, or about 11 times as much variation in every 182.62 days of elliptical orbit.

For the most part, the Earth's orbital velocity isn't that relevant. The Earth's position, not it's velocity, determines which stars or planets you see at night and if the Earth's spans slightly more or slightly less of it's orbit than usual, over a 24 hour period because of the Moon, 5 minutes of fluctuation over 13 days is only relevant to the most rigorous of astronomers.

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