I've come to believe that in Kirk's time the Enterprise wasn't always orbiting the planet, but was instead hovering on it's impulse engines above a fixed point on the planet's surface. For example Assignment: Earth, the Enterprise was stationary two hundred miles directly above the New York City. At that altitude she would have been orbiting at five miles a second, but at a hover she is moving forward at only at a third of a mile per second, matching the city below her. Impulse engines that could propel the ship at hundreds of gees through space can easily provide one continuous gee to hover.
Nothing is more subtle than two thermonuclear engines spitting out enough hot reaction mass to keep a million ton vessel stationary over a major metropolitan area during the Cold War.