Reaching orbit

Space is only about 100 km away, closer than many people’s commute. Getting up there is easy: sounding rockets do it with a few minutes of flight and fall straight back. Staying up there is the hard part. To stay you need about 7.8 km/s of sideways speed, and almost all of a rocket’s fuel is spent on that.

Going up takes minutes. Going sideways fast enough takes almost all the fuel.

Look

A launch has four acts:

1. Straight up for a few seconds, to clear the tower and the thickest air.
2. The gravity turn. A small tilt, a couple of degrees, and gravity does the rest: the path bends over by itself while the engines push along it. No sharp turns, which would break a thin rocket in the air stream.
3. Staging. After about two and a half minutes the first stage is empty and falls away. The upper stage, much lighter, keeps pushing.
4. Flying almost flat, building sideways speed until it reaches orbital speed. The engine shuts off. The spacecraft is in orbit, circling the Earth every 88 minutes or so.

Launching eastwards helps: Earth’s own spin gives a free 0.465 km/s at the equator. That is why most launch sites face an ocean to the east.

Understand

In the simulation above, the vehicle spends about 9.3 km/s of “delta-v” (the total change of velocity its engines can provide) to reach a speed of 7.8 km/s at 200 km. The difference is gravity loss: while the rocket still points partly upwards, part of its thrust only holds it up against gravity instead of speeding it up.

Real rockets also fight air drag and lose some efficiency to steering. The usual budget quoted for low Earth orbit is 9.3 to 10 km/s, which matches.

Moment in the simulation	Time	Altitude	Speed
First-stage engine cut-off	about 2 min 24 s	about 90 km	about 3.0 km/s
Second-stage cut-off, in orbit	about 7 min 37 s	about 200 km	7.8 km/s

The first stage, cut off at 3 km/s, is far too slow for orbit and falls back (Falcon 9 boosters then steer themselves back to land).

Master

The upper stage in this model uses a simple closed-loop guidance law. At each moment it chooses a pitch angle θ so that the vertical acceleration corrects altitude and climb rate:

a_T sin θ = μ/r² − v_h²/r + k₁(r_target − r) − k₂·v_r

The first two terms are gravity minus the “centrifugal” relief of horizontal speed. As v_h approaches the circular speed √(μ/r), those two cancel and the stage points flat. Real vehicles use more refined schemes (for example the Apollo-era iterative guidance mode, or powered explicit guidance on the Space Shuttle) that minimise propellant, but the principle is the same.

Inserting into orbit is ending with zero vertical speed exactly at the target height, and horizontal speed equal to √(μ/r). Miss either and the orbit is an ellipse, with its low point possibly inside the atmosphere.