# orbital speed to maintain relative position to the Earth

I note that the orbital speeds of the planets slow down with increasing distance from the Sun. Say we have a space station orbiting the Sun at a distance of about 400 million kilometres (within the asteroid belt) in a circular orbit. Say the satellite is diametrically opposed to the Earth, ie, 'hidden' behind the sun. Is it physically possible to maintain the satellite's position relative to the Earth, making one revolution of the Sun in one Earth year, even though it is much further out and would have to move a lot faster than the Earth?

• Not without constantly thrusting which is probably impossible with current technology. Commented Feb 6, 2023 at 21:56
• Does it have to be in the asteroid belt? If you brought it in closer, say to Sun-Earth L3 (directly opposite from Earth at about the same distance - just slightly closer), you'd achieve basically this. You would need some minimal station-keeping thrusters, as L3 isn't very stable. But if you're looking for an orbit that won't be visible from Earth, it's either that or maybe Earth-Moon L2 (far side of the Moon). Same stability issues apply there. Commented Feb 7, 2023 at 18:04

No, not without a rather large acceleration constantly pushing it towards the Sun.

The circular orbital velocity is: ($$\mu$$: mass of the Sun times the gravitational constant, $$r$$: distance from the Sun).

$$v_c = \sqrt{\frac{\mu}{r}}$$

As you observed, diminishing as you go further out.

To have the same orbital period as Earth though, the velocity scales linearly with distance, so at 400 million kilometres, it has to go 2.67 times faster than the Earth, while the circular velocity is 0.61 times that of Earth.

So it is going 4.35 times faster that it need to in order to stay in a circular orbit! And violently so. Even at $$\sqrt{2}$$ times circular velocity objects get thrown out of the solar system (the escape velocity)

$$v_e = \sqrt{2} \cdot v_c$$

Providing the required acceleration isn't a theoretical impossibility though. A rocket engine could burn to push it towards the Sun (quickly running out of propellant), or a piece of string attached to the Sun could physically restrain it (requiring improbably material strength). For the reverse problem, keeping altitude while orbiting too slowly, a not so far out idea is to use the radiation pressure from the Sun to push against a Solar sail to provide acceleration.

The acceleration required can be calculated from the angular velocity $$\omega$$

$$a = \omega ^2 \cdot r - \frac{\mu}{r^2}$$

Here ~1.5cm/s²

• "keeping altitude while orbiting too slowly" Technically, my quadcopter does that... :D Commented Feb 6, 2023 at 22:32
• Could you add how many Gs this station would experience to keep it in 'orbit'? Commented Feb 7, 2023 at 14:19
• Hey, thanks for all input. My query has been answered. Lagrange points sound a lot more feasible. Commented Feb 8, 2023 at 1:29
• @user50566 welcome to the site! Since you're new here, I'll point out that the method for indicating that an answer is correct / met your needs is to click the gray checkmark beside it. Commented Feb 10, 2023 at 16:40
• Thanks Organic Marble. Answer checked😊 Commented Feb 11, 2023 at 21:40