How do i calculate the Delta V from surface to the minimun orbit possible

[Askert]

Guys, i was trying to learn things about ksp and i dont know how do i calculate the delta V necessary to make a orbit from surface, i tried to substitute the formulas already knowed and i've got nothing productive. As i know that exist a image telling about the delta v costs, i just want to know how to calculate it and try my self with math. im new, sorry.

[Deddly]

Hi @Askert, it looks like you accidentally posted this in the Portuguese section, so I've moved it here to Gameplay Questions in English.

[HebaruSan]

There are three main areas in which you need to spend Δv to get to orbit:

1. Orbital velocity
   This is simply how fast you must be going once you're in orbit to stay in orbit. The vis-viva equation is great for calculating this (you can simplify it a bit for circular orbits):
   https://en.wikipedia.org/wiki/Vis-viva_equation
   You'll need to know the mass of the body from which you're launching (M), which can be gotten from the KSP wiki:
   https://wiki.kerbalspaceprogram.com/wiki/Kerbin
   ... and the size of your orbit (r and a), which is the body's radius (also available from the wiki) plus your altitude in orbit (generally 70 km or thereabouts for "the minimum orbit possible").
2. Gravity losses
   To get to orbit, you have to travel at least 70 km upwards. That costs energy, and the whole time you're climbing up there gravity will be pulling you back down. Unfortunately rather than something like the vis-viva equation, gravity losses require solving complicated differential equations that relate to your exact flight path, so folks tend to determine this by trial and error, generally following approximately optimal paths to orbit to control for many of the variables.
3. Drag losses
   When launching from a body with an atmosphere, gravity is not the only force holding you back; you also need to spend extra energy to punch through all that air between you and space. Much like with gravity losses, this is very complicated to pre-calculate, but unlike gravity losses it also depends on the thickness of the atmosphere at different altitudes and the details of how your craft is put together; it'll take a lot more energy to push a big draggy sail through the atmosphere than a sleek spear. Again people usually apply rules of thumb to avoid overly draggy craft and then use trial and error.

If you're curious to see a code implementation of a mediocre (but fast) approximation of this calulation, I had to write one for Astrogator. It approximates the difficult components with hard coded linear factors determined empirically using a spreadsheet of the Δv requirements and mass and atmospheric data. I wouldn't necessarily recommend using this exact calculation yourself, but it can give an idea of what's involved and why there isn't one simple equation you can use:

https://github.com/HebaruSan/Astrogator/blob/a57c8ec15a4137abbd5c2b5fe31127531173a136/Source/PhysicsTools.cs#L63-L95

EDIT: This link from that code is a previous discussion covering most of the same concepts better than I have here:

Edited May 18, 2022 by HebaruSan