Kosmo-not

Gravity turn is the best way to go. Just pitch over a little bit after liftoff (how much depends on the TWR you have) and then don't touch the controls except for throttle.

The Oberth Effect simply states that the power a rocket produces is higher at higher velocities. power = energy / time energy = force * distance (also referred to as work) For example, assume a rocket engine burn for some arbitrary amount of time. At a higher velocity, the rocket travels through more distance for the same amount of time. Same force applied over a larger distance equates to more work energy done. More work energy done over the same amount of time equates to more power produced. Applied to orbital mechanics, energy is most efficiently added/subtracted from your total orbital energy (kinetic energy + potential energy) at the lowest point in your orbit, where you are travelling the fastest. That's all there is to it.

I decide to log into the KSP forums after many many months of inactivity, and it's @Vanamonde who wants to pull me out of retirement! I'll go ahead and try to clean up the original post the best I can. Assuming you can create maneuver nodes, there is a much MUCH easier way to accomplish an interplanetary trajectory: 1.) While orbiting Kerbin (I use a probe for this), create a maneuver which will take you juste barely out of Kerbin's sphere of influence (theoretically, you're exiting the SOI with zero velocity, achieving the same orbit as Kerbin around the sun). 2.) From the orbit around the sun, create a maneuver for a transfer orbit to whatever you want to intercept. Manipulate it until you get an intersection. 3.) Timewarp until Kerbin is at the maneuver node created in step 2. 4.) With the craft you are going to be doing the stuff with the other planet you want to go to, create a maneuver node for an ejection from Kerbin's orbit (in either the planet's prograde or retrograde direction) and increase the velocity until an intersection is made with the target planet. You will have to adjust in the normal or anti-normal direction for planetary orbits which do not lie on the origin planet's orbital plane. 5.) Execute maneuver and later do whatever slight corrections are required. Any such undertakings are at your own risk. Upon attempting the above steps, you agree to not hold Kosmo-not liable for any damages resulting in the loss or destruction of spacecraft, kerbals, or ego. *update* I think all the equations in the OP are fixed now.

Have you tried a rocket design?

Is there somehow a problem with orbiting that low?

The Apollo lunar ascent module was an SSTO. An SSTO can also be used to reduce complexity and thus less things to go wrong. The reason for using an SSTO is really up to the designer.

The point of an SSTO (Single Stage To Orbit) is to get into orbit with a single stage. What you do with that stage beyond that is up to you. I am able to recover the first stage of my rocket, so . Sure it reduces the overall costs of a mission with that rocket by about 10k (compared to launching without recovery equipment), but is it really worth it to me? I don't know.

I don't have any problems with reentry with my SSTO. I separate the command pod and use that. It doesn't even need a heatshield from LKO. For any of my Mun missions, I add a heatshield with no ablator; does the job just fine.

I wouldn't really consider that a challenge. It's very easy to design an SSTO as long as you don't overdo the upper stages. I would say a very low TWR lander (1.5 to 2.0 with respect to Mun) with a tight delta-v budget is a good test of skill.

Try this SSTO

Here is a picture of one of my early career SSTOs: Very easy to fly and get into orbit. So to answer your question: Yes, early career SSTOs are easily possible.

Try putting the rear landing gear close to the center of mass. It looks like it's currently forward of most of the control surfaces on the wing, so pulling back on the stick is actually pushing the plane into the ground. You can also try putting canards on it near the nose.

The Goddard Dilemma only applies when there is an atmosphere, and is only used for ascents. The whole point of the gravity turn is to keep the rocket pointed into the relative wind. A Mun lander I flew about 2 years ago started the landing burn with a TWR of 1.09 relative to Mun. I heard from someone that doing a gravity turn landing (only pointing at retrograde) required the craft to start at 30km altitude. It also used a lot more fuel than coming in low and killing horizontal velocity. This is very apparent in low TWR craft. That said, you can get away with any sort of landing method if you have a high TWR. Here's the video before someone else posts it:

The most efficient way is to come in horizontally, close to the ground. Kill off your horizontal velocity while adjusting pitch to control your vertical speed. It's very easy to do.