A Primer on Ascent: Your Periapsis and You - KSP-RO/RP-1 GitHub Wiki

Required background reading: Myths 1 and 5 of False KSP Lessons

TL;DR available at the end. That said, reading this entire page is strongly recommended.

0. Pre-Introduction

This article deals with the case where a high TWR / low burntime rocket burns down towards the end of a normal orbital insertion at a pitch angle of 20 degrees or so which is entirely normal. There are at least four issues which can lead to rockets burning straight down radially at the Earth which indicate you have larger problems than what this wiki page describes:

Not having a sufficiently aggressive pitch program (Pitch Start is too high and/or Pitch Rate is too low) where by the time the rocket has gone through maxQ it is at a very high pitch angle and has difficulty pointing horizontally fast enough and the Apoapsis raises up too high. This is something that is controlled and is fixable by the user. Generally increase the Pitch Rate, if you increase it to 2 degrees/second try dropping the Pitch Start to 25 m/s. Read the documentation on PVG.
Using a coast phase with residuals. As of today (March 9th, 2023) the version of PVG that is available on CKAN does not handle residuals with a coast well and the booster stage will burn and continue to raise the apoapsis, which can lead to the later stages needing to burn down hard to achieve the orbital insertion parameters. This will appear to be random based on how long the residuals burn compared to the prediction. The fix is to set a Fixed Coast Length of 0 seconds.
Insufficient Avionics. This will show up in the PVG status window in ALL CAPS when it happens, and the usual message about insufficient avionics will flash across the screen, generally upon staging into the deficient stage. The rocket will act like an unguided sounding rocket until it burns below the avionics limit and becomes controllable again. This will again push the Apoapsis above the insertion altitude. The fix is to rebuild the rocket with correct avionics.
Apoapsis Attachment. Sometimes PVG will attempt to attach at the Apoapsis of the target orbit instead of the Periapsis. This has all the problems that have been highlighted below but it is due to the optimizer finding a valid, but unwanted solution, with a high altitude. The solution to that is to use "Attach Altitude" to force it to attach to the Periapsis. In more current versions of PVG this option has been renamed "Burnout Altitude" and in those versions the optimizer should avoid apoapsis attachment.

A. Introduction

Learning to fly ascents in RO/RP-1 can be hard. If you've got enough delta V for a 2500 x 150km orbit, surely you have enough for 400 x 400km, right? It's not so simple. And why is PVG taking your rocket to a far higher apoapsis than you asked for? To get into why, first we have to talk about what the Apoapsis and Periapsis of an orbit actually are.

B. The Apoapsis and Periapsis

If you've played KSP at all before RO/RP-1, you've run into these. They're the highest and lowest points of your orbit. But from that simple definition comes a few other details you may or may not realize.

You are moving fastest horizontally at your periapsis. You might have heard about how that's the best time to burn because of the Oberth effect.
You are moving slowest horizontally at your apoapsis. You might have heard that that's the best time to alter your orbital plane.

But note I said "horizontally" there, not fastest overall. That's because your apoapsis and periapsis have another feature, and this one they share in common: zero vertical velocity. This makes sense when you think about it: on your way from periapsis to apoapsis, you're climbing, and once you get there--for one instant--you are neither climbing nor falling back down...and then you start heading back to your periapsis and start heading down again. The process repeats itself in reverse at periapsis.

Now, you may ask: what relevance does this have to flying ascents?

C. Orbital Insertion

As we saw in B above, there are only two points in your orbit where you have zero vertical velocity: your apoapsis and your periapsis. Let's turn that statement around: if you are not at zero vertical velocity, then you are (a) higher than your periapsis, and (b) lower than your apoapsis. That's by definition; if you have positive vertical velocity, then your periapsis is by definition behind you and your apoapsis in front, and if you have negative vertical velocity, your apoapsis is behind you and your periapsis is in front of you.

This matters for ascents because it means that the orbit you get from your ascent is a product of three things:

The altitude at which you stop your ascent burn.
Your horizontal velocity at burnout.
Your vertical velocity at burnout.

Ordinarily we'd combine 2 and 3 and call it your velocity vector, but I'm very intentionally leaving them separate in this case.

If you're coming from KSP, you're used to an ascent that involves burning upwards to establish an apoapsis and then circularizing ("inserting" -- circularizing is a special case of orbital insertion where the resultant orbit has a very low eccentricity) at that apoapsis. You're already subconsciously used to the implications of B above, that to establish a circular orbit you need to do so at apoapsis (or at periapsis, if you're circularizing around the ~~Mun~~Moon). But let's generalize that rule: if you want to insert at either periapsis or apoapsis, you have to have zero vertical velocity at the end of your insertion. But you also have to be at the altitude you want for your periapsis (or apoapsis).

D. Burn Time (Your Periapsis and You)

We've seen, in B and C above, that in order to insert at periapsis (the common case for an ascent--either you're going to a circular orbit, or you're going to an eccentric orbit and it's much cheaper to insert at periapsis than apoapsis due to lower gravity losses) that you need to reach the altitude of your desired periapsis, and the horizontal velocity required for your orbit (i.e. your SMA, the combination of your periapsis and apoapsis), and lastly you need to reach that point at or very near zero vertical velocity.

Modern launch vehicles, that take 10+ minutes to get to orbit, have great flexibility here. But early launch vehicles--things like Thor-Able, or Atlas, or R-7--well, they only have about 5 minutes of burn time. Why does burn time matter? Well, if you have a burn time of 5 minutes, you need to:

reach your periapsis altitude, which means
burning up a lot so you get there in time, but also
burning down a lot near the end of the burn so you cancel all that excess vertical velocity that got you there fast.

And during all of that also build up all the horizontal velocity you need for orbit.

Now, gravity helps you out here: it decelerates you vertically, aiding in (3). For a rocket with a long burn time, it helps you out so much you never have to burn downwards, but for those early launch vehicles you'll have to burn down some even with a super-low periapsis like 150km. Using a coast with an early rocket is effectively extending the time allowed to get to that target altitude, therefore removing component (3) and lower the speed needed to be applied in (2).

When you think about it like that, it becomes clear why you need to use a low periapsis (or a long coast) with an early rocket: it's incredibly expensive to burn upwards so fast that you reach, say, 300km before 5 minutes are up and also burn downwards (slowing your vertical velocity on your way to 300km!) such that you end up at zero vertical velocity at 300km. Because the more delta V you spend burning upwards, the more delta V you'll have to spend later to cancel out (which means the longer you'll spend canceling it out, which means more time at slower-than-maximum vertical velocity, which needs even more upwards burning, etc.)

E. Why Is PVG Burning Past My Apoapsis / Why Is PVG Burning Down?

Given D above, the answer should be pretty clear. When you tell PVG to try to achieve a given orbit, it does exactly what you ask (assuming that's possible at all--a high enough periapsis won't be). Let's say you ask for 400 x 400km. If you have an early rocket (with a ~5 minute burn time), here's how that has to work: PVG has to make sure that, at the end of those 5 minutes, you are at 400km altitude, ~7.8km/sec orbital velocity, and 0 m/s vertical velocity. How it achieves that is by burning up lots (you will see a reported apoapsis well above 400km before it stops burning upward), and then burning down lots (apoapsis slowly lowers back down to 400km, and should hit 400km just as your rocket finishes ascending to 400km), all while burning sidewards lots (to get you that 7.8km/sec orbital velocity).

F. Ok, That's Cool, So What Do I Do?

You have three choices.

Design a different launch vehicle. We're going to discard that, because it's not a real solution, and it's especially not an optimal solution (a direct ascent to a 400 x 400km orbit is way more expensive than an ascent to a parking orbit).
Coast. This is an acceptable solution if you have a rocket design that can do so (it requires guidance and ullage and RCS for orientation, all on the coasting stage), but it is again not an optimal solution at all, because coasts lead to some pretty big gravity losses.
Launch to a parking orbit and then raise periapsis later.

(3) is the preferred option, because it's the cheapest in terms of delta V. However, it requires either an extra stage or a relightable engine, and requires you to either not care too much about the final periapsis (say, an early Comsat contract where the requirement is just Ap above x value, Pe above y value) or have full guidance on that extra stage (or the relightable engine, again). Here's how it works. You tell PVG to launch to 150km as your periapsis and (whatever apoapsis you want). Then you make a maneuver node at apoapsis to change periapsis to whatever you want. If your last stage is unguided, aim at the node, spin up, and decouple the upper stage; if you have a relightable engine or a guided last stage, just execute the maneuver normally. Congratulations, you've saved hundreds (or, if high enough, thousands) of meters per second of delta V!

G. The TL;DR Edition

In one sentence: It’s not just about raw delta-V, it’s also about burn time.

Why?

When you ask MechJeb to insert you into an orbit with some given periapsis, you’re in fact asking for several things:

When you reach your requested periapsis, you are at orbital (i.e., horizontal) velocity.
When you reach your requested periapsis, you have no vertical velocity. This is what makes your periapsis an apsis, by definition: It’s the lowest point on the orbit, so your vertical motion reverses (going down → going up) as you pass through it. This means:
1. First, you need to accelerate up to get to your periapsis altitude.
2. Then, you need to accelerate down so that you stop going up once you get there.

This point (2) is not necessarily immediately obvious and is why (a) MJ sends you to a higher apoapsis than you requested and then burns down; and (b) you can get to a 200x1500 km orbit but can’t get to a 400x400 km orbit.

Points (2.i) and (2.ii) both take time and energy. Think of it like an odd foot race where you need to be stopped at the finishing line: It takes a lot more effort to sprint and then slow down than to walk. An early LV needs to sprint to its periapsis, because that’s all the burn time it’s got (~5 minutes). This means expending a ludicrous amount of energy to go up — a high apoapsis — before expending an equally ludicrous amount of energy to slow down at the end — burning down. And at some point, the energy requirement becomes so ludicrous that your LV simply can’t make it.