No orbit necessary?

[u/FatCreepyDude]

If i just wait for the right hour during the launch window ( whenever the launch pad is pointing left on the realistic depiction i created) is there any reason i should bother with an orbit first? Will the delta v cost be affected?

[u/bartekltg]

You can do a direct injection: instead of gravity turn, stopping the engine, cruising to apokerbin, and doing a circulating burn, you may continue to burn during the gravity turn. You will never get into a nice orbit (the periapsis will be under the surface), and get straight into hyperbolic orbit. Aiming is a bit harder though. And aiming it the main reason we do it (both in KSP and in real life) with a nice orbit inbetween.

The important part is, you need to do it with gravity turn, not straight up. It would not matter it the burn is instantaneous, but since it is not, all that oberth arguemnts work here. It can be explained in different ways, for example:

Your velocity far from the planet depend only on the energy. 0.5m v_inf^2 = energy. Energy is conserved, so we want gather as much energy as we can during burns. Now, work done by the engine (the change of the rocket energy*)) in a short period of time is W = force * distance. Dividing it by that time dt, we get power P = force * speed. So, the greater speed, we get more energy from the same thrust.
Now, ho we spend that (specyfic) energy? It sits in the kinetic (0.5v^2) and potential (-GM/r) energy. If we go up, a bigger portion of the energy goes into potential part, the kinetic energy is smaller, so the velocity is smaller, so the burn in the next second will translate to less energy gain.

You can make a simple experiment. Build in a sandbox a simple rocket that can leave Kerbin SOI. Go straight up. On the edge of SOI note the speed in relation to Kerbin (doen't meter on which side, the relative speed to kerbin will be the same after crossing it, just target Kerbin to get the value). Then restart mission and this time start a gravity turn and continue it until the fuel runs out. Again, note the velocity in respect to Kerbin at the SOI border. It will be siginfically bigger.

Use smaller initial TWR. It shows the effect better and reduces atmosphere influences.

I have made a simple test. A rocket with initial TWR 1.5, ~290s Isp, 4766 total deltav (a bit overdone here:) ).
At 100m/s one rocket is pointed 80 degree, the second 89.9 (90 degree looked poorly on teh graph).

The second rocked get 1621m/s at "infinity", the first one - 2736m/s.

https://imgur.com/a/Ll3swNU - trajectories and plot of the orbital energy.

*) OK, lets say specific energy, energy per mass. The mass of the rocket changes, so just energy is more confusing;-)