How it is calculated
The radius used is a mean radius, and the atmosphere does not drag. A real orbit sits above the surface, so circular speed there is lower.
v1 = √(GM / R), v2 = v1 × √2
Where the numbers come from
- Body
Example
Earth's first cosmic speed is about 7,91 km/s.
Questions
Why not exactly 7,9?
It depends on the GM and radius that are used. Here Earth’s GM is 3,986004418×10¹⁴ m³/s² and the radius is 6 371 km.
A satellite in a 400 km orbit?
This screen’s formula will not take the larger radius for you: R is built into the body.
The Sun?
It is not on the list. Weight on the planets has its own calculator. These are different tasks.