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Gravitational constant: 6e-11 (meters, seconds, and kg).
Jupiter radius: ~ 69,000,000 meters (google)
Jupiter mass: $2*10^{27}$.
1 kg, at Jupiter's radius, experiences ~100N of force:
$$ F = G * \frac{J_m * 1.0}{r^2} \approx 6*10^{-11}\frac{2*10^{27}}{5*10^{15}}\approx 20_N $$
However, if one just moves out to 100,000 kilometers, you get down to $\approx 10_N$, the same as earth.
What explanation is there for why there is no atmosphere at the earth-equivalent level of gravity?
Perhaps it is simpler not to compare Jupiter to Earth, but to think about what happens as you add more and more gas to a gas planet. Even if you make a planet out of incompressible stuff, the gravitational field at the surface increases as the planet grows. This effect is magnified when making a planet out of compressible stuff (gas). The pressure inside the planet goes up for two reasons: there is more mass pressing down from above, and more gravity pulling on it. So the density increases, meaning that the volume of the planet grows more slowly than its mass. The radius of a planet made of incompressible stuff would grow like $M^{1/3}$. We've just decided that the radius of a gas planet grows more slowly. Meanwhile, the radius at which the gravitational field of the planet reaches some small value (small so as to ensure it is outside the planet) grows like $M^{1/2}$ (that is, faster than the radius of an incompressible planet, and even more faster than that of a compressible one). Thus it is clear that as you add gas, you steadily increase the gravitational field at the "outer edge" of the planet, however you choose to define that.
the fact that there is no atmosphere near jupiter at the equivalent g level radius for earth is because any parcel of gas at that radius has no solid surface to stop it from falling closer to jupiter. so it falls, and hence there is no appreciable amount of gas at that distance from jupiter.
The strength of gravity at a point is not directly related to the pressure of the atmosphere at that point. Instead it is related to the pressure gradient. To a decent approximation
$$ g = \frac{1}{\rho}\frac{dP}{dz} \sim \frac{T}{P}\frac{dP}{dz} $$
as this is the pressure gradient that provides an upward force to cancel out the downward force of gravity.
Even ignoring the solid surface of the earth or temperature differences, solving this gives different surface pressures for planets of the same surface gravity but different sizes.
