# Gravity - change with altitude

Does acceleration due to gravity, $g$, which is calculated by Newton's equation, decrease with altitude? Forgive the silly question.

• Yes, it does, but it does it only slowly. The acceleration of gravity will be a quarter of its value on the ground at an altitude of one Earth diameter. Even at 10000m, as high as passenger jets go, it has only decreased by a fraction of a percent. May 26, 2015 at 3:23
• I am afraid your question is a little vague. I believe you are asking if the acceleration decreases with an increase in altitude, then the answer is yes. If you however mean that the acceleration decreases with a decrease in altitude, study more. May 26, 2015 at 8:31
– rob
May 26, 2015 at 12:50
• @rob I agree I may have been too harsh. I apologize. May 30, 2015 at 13:22

The gravitational force on a small mass $m$ some distance $R$ from the center of a large spherical mass $M$ is given by $$|F| = \frac{GMm}{R^2}.$$ If your distance from the center is some altitude $r$ above the radius of the Earth's surface $R_\oplus$, the force is $$|F| = \frac{GMm}{(R_\oplus + r)^2} = \frac{GMm}{R_\oplus^2} \left( 1 + \frac{r}{R_\oplus} \right)^{-2} \approx \frac{GMm}{R_\oplus^2} \left( 1 - \frac12 \frac{r}{R_\oplus} \right)$$ So for $r\ll R_\oplus$, you can say that the gravitational force gets weaker linearly with altitude. Low-earth orbit has $r/R_\oplus \lesssim 25\%$, so for higher "low" orbits this approximation starts to fail (you could take the next term in the binomial expansion if you wanted). The atmosphere has $r/R_\oplus \lesssim 1\%$, so all gravitational accelerations in the atmosphere are the same to about three significant figures.