Why is my third cosmic velocity calculation giving 13 km/s instead of 16 km/s?

[Heirot]

... is the minimal speed (relative to the Earth) which must be imparted on a body resting on the Earth's surface to escape the Solar (and Terrestrial) gravity. So, if one uses the conservation of energy (neglecting the Earth's rotation about its axis), one has

1/2 m (v_3 + v_o)^2 = G m (M_s / R_s + M_e / R_e), where

m - mass of the body
M_s/e, R_s/e - mass of the Sun (Earth) and distance from its center to the body
v_3 - third cosmic velocity
v_o - the speed at which the Earth orbits the Sun
G - gravitational constant

Now, this equation gives v3 = 13 km/s, while all the sources cite v3 = 16 km/s. Am I missing something?

Thanks

 

[Meir Achuz]

Shouldn't your R_s be the distance of the Earth from the Sun?

 

[Heirot]

Yes, R_s is the distance form the Sun to the Earth. Anyhow, all other refrences give completely different expression than mine (e.g. Irodov - problem circa 1.230). I think that my energy calculation is ok. Or am I missing something?

 

[Meir Achuz]

The sum v_o+v_3 should be a vector sum. Your answer is for the two velocities parallel.
There is a range of answers for v_3, depending on the angle between them.