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I currently want to achieve something that I have in mind. I want to "theoretically"(only the calculation part) be able to make a Lunar Mission by either sending a Satellite to the Moon, or landing a Rocket on the Moon. In order to do that, I should first get the Rocket into an Earth Orbit. Getting it into Orbit requires a Horizontal Velocity which can be calculated from the VisViva Equation: $$\vec{V}=\sqrt{\mu (\frac{2}{r}-\frac{1}{a})}$$ So we are able to get it in Orbit. Let us calculate the Burn Time too, to know for how much Time the Thrusters should be on. We can do that from the Tsiolkovsky Equation, but that's another story(which I have already solved). There are more steps we should do, but a major one is missing. Gravity Turn. In order to be efficient and make the mission not be that expensive, we should perform a Gravity Turn. I have read from my Book(Fundamentals Of Astrodynamics) that most Rockets usually start the pitch programme 15s after the lift-off. But how do we calculate that? These are my questions: How do we calculate when to start the pitch program? How do we calculate the Angle of the Gimbal Thrusters we should have at the pitch program? How do we calculate for how long to have the pitch program on before aligning the Thrusters with the Rocket's Nose? To be honest...I don't really know. So let me rephrase analytically. I want to learn the Equations that are being used by real Space Agencies for real Space Missions. How do we calculate when to start the pitch program? What should the Angle of the Gimbals be? For how long should the program keep running until we turn it off after some Time to let Gravity alone tilt our Rocket to the East for the rest of the Flight until we reach the altitude we want to get our Rocket into Orbit?

ATTENTION: I do not want solutions of Numerical Integration of The N-Body Problem, as I could already solve it. I want Patched Conic Approximation. I want that because I want to be able to solve them the "old school" way by paper and pen, not start Programming an entire Programming to Numerically solve it for me. If you did not understand something that I said tell me. Do not be shy to tell me as I am really interested in that, really locked in and focused and really serious. You can also check a post that I made at Reddit that is about the same question:https://www.reddit.com/r/AerospaceEngineering/comments/1paq35i/math_behind_gravity_turn_in_depth/ I would REALLY APPRECIATE it if you helped me. By the way I am from Greece and I am 15 Years Old. You can also check this image of a drawing that I made showing a Rocket performing a Gravity Turn[!errorenter image description here]1

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Nick Pap is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
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    $\begingroup$ “I want to learn the Equations that are being used by real Space Agencies for real Space Missions” — There are a lot of such equations. Your endeavor is noble but I fear that this needs more focus as a question. Maybe pick one particular point of research (e.g. calculating patched conics) and then leave the rest (landing zone calculations, delta-v calculations, ship maneuvering, etc.) for other questions. I may also recommend you to Space Exploration Stack Exchange. $\endgroup$ Commented 10 hours ago
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    $\begingroup$ It isn't what you asked, but you might find these Kerbal Space Program Tutorials by Scott Manley helpful. This is a video game that can teach you real physics about space flight. You might start with the 3rd one, Kerbal Space Program 101 - Tutorial For Beginners - Construction, Piloting, Orbiting $\endgroup$ Commented 8 hours ago

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Without an atmosphere you could pitch over immediately, turning to the shallowest angle you can that still causes your vertical velocity to increase. The reason for the slow pitch over is that the lower atmosphere fights you, its drag slowing you down and making you expend more energy to reach the same velocity compared to if you were above the atmosphere. Since you know you need a certain horizontal velocity to get to orbit, then going up for a bit to make your burn more effective makes sense.

The specific length of the delay is not calculated exactly. I believe agencies get it from computer simulations of the rocket in the atmosphere, testing different trajectories and settling on the most efficient one. It's a balancing act between fuel spent on altitude and fuel saved on drag.

If this interests you, then simulating an ascent and finding the best parameters could be a cool project, Here is someone that did that and posted it on YouTube. Alternatively Kerbal Space Program could also teach you a lot about the subject, especially with trajectory planning mods such as MechJeb and KOS.

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sagiksp is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
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