Is an orbital launch more like a vertical climb or a Hohmann Transfer?

Question

It is said that a launch to orbit requires a Delta V of 9.3–10 km/s, and this encompasses orbital speed, air drag, gravity drag, and the energy to raise altitude to low Earth orbit. But I'm troubled by vastly different estimates of the amount of Delta V that goes to the altitude term.

To demonstrate, consider the two obvious physical models and the numbers they produce. First, we have a perfectly vertical "toss" of a rocket, like a sounding rocket or a suborbital flight. Second, we can ignore Earth's atmosphere and imagine two burns to reach LEO. Assuming 200 km orbit:

1. $v\approx \sqrt{2 h a} $, for about 1.98 km/s, or use the more accurate $GM/r$ potential for 1.95 km/s
2. With the Hohmann Transfer formula, get $\Delta v_1$ and $\Delta v_2$ to be 61 m/s and 60 m/s, for a grand total of 121 m/s, or 0.121 km/s

That's a big difference there. For an actual rocket launch, what is the appropriate mental model? On the one hand, the early parts of the launch aren't very close to the horizontal. But on the other hand, the characteristic height of our atmosphere is like 8 km, which is much lower than the altitude of the orbit.

If one were to give a ballpark figure for how much of the delta v budget goes into raising the altitude, what would be a good estimate?

Furthermore, does the burn even last for half an orbit? How Hohmann-ish is the climb to orbit?

Answer 1

Is an orbital launch more like a vertical climb or a Hohmann Transfer?

This is a false dilemma. Neither is correct. Both of your calculations are wrong.

Your vertical climb interpretation ignores that the velocity at the end of the launch needs to be horizontal and about 7.8 km/s. Your Hohmann transfer interpretation ignores that the velocity at the start of the launch is not a horizontal velocity of about 7.8 km/s.

Assuming a non-rotating, atmosphere-free Earth and impulsive delta Vs, your vertical climb can be fixed by adding a 7.784 km/s delta V on reaching the maximum height. This gives a total delta V of 9.733 km/s and an elapsed time of a bit over 4 minutes. Your Hohmann transfer can be fixed by adding a 7.905 km/s delta V to your initial impulsive burn. This gives a total delta V of 8.027 km/s and an elapsed time of a bit less than 90 minutes.

The problem with both of the above calculations are the assumptions of a non-rotating, atmosphere-free Earth and impulsive delta Vs. A real launch takes about 8 to 11 minutes to complete. Thrusting is finite and more or less continuous, and the trajectory gradually turns from purely vertical at launch to purely horizontal at orbit insertion.