Eagle1Division
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Hey,
For a long time I've been wanting to be able to work calculations by hand for all the maneuvers conducted in orbiter, for it's own sake, but mostly to design realistic vehicles for Orbiter. I'm just getting into Calculus as a senior in high school, so lots of things are starting to open up. Recently I've discovered capital-sigma notation (aka summation notation), for instance (Well, on my own. So I don't really know it well...).
But one of the big mysteries I'm trying to get at right now is ascent trajectory, specifically the pitch program, how to mathematically solve to find the most efficient pitch program.
That's a long ways off... But for now I think I may have made one step, at least I know how to calculate gravitational drag! (I think)
I tested this in orbiter, and it was somewhat accurate, but for some reason there was an amount of error, that I'm thinking it may be because of the accuracy of the calculator I'm using (TI-84 Plus, it's trig isn't perfect)...
G = Cos(90/Ov * V) * GM/R^2
Cosine done in degrees.
G is the negative vertical acceleration due to gravity, in m/s^2.
Ov is orbital velocity
V is the vehicle's current horizontal velocity
And GM/R^2 is the equation for the force of gravity.
Is this correct? At least it seems to be, but my calculator and Orbiter aren't agreeing completely. To test it I took an ascent in the DG-S, and cut off the engines at different points to measure vertical acceleration due to gravity for different speeds. My error was usually in-between ~0.02 all the way up to ~0.1 m/s^2.
So to find the total gravitational drag, I simply plug this into summation notation (Which I don't know how to type on the forums :-/ ), with V as my changing variable, assuming a constant rate of horizontal acceleration...
Which is a problem. Since the rate of horizontal acceleration will depend on pitch and fuel used of the vehicle, it isn't anything near constant. So I still have no idea on how to find the most efficient pitch program.
And for the second part of my post:
I've been trying to find encounter velocity for orbital transfers. I need this for Delta-vee calculations of interplanetary and translunar vehicles. So what I did, is this:
Vapp = Sqrt(Vesc^2 * Vperi^2)
Vapp = Approach velocity
Vesc = Escape velocity at closest approach
Vperi = Velocity at closest approach (apogee of elliptical orbit around major body. In the case of TLI, Earth)
In TLI, I found I was about 200 m/s off, but it's hard to tell how accurate my math was, some of it may have been non-spherical gravity sources and course correction burns, seeing as TransX isn't perfect, but 200 m/s isn't really acceptable error for designing a Trans-Lunar Vehicle! And I don't even know if that equation actually works or not! Though it seems to, to some extant.
As exciting as the capital sigma (aka summation) notation was, I don't think it solves this problem of encounter velocity. Since as you fall towards a body, it's gravity will cause you to accelerate at a rate dependent on your distance, and your distance is dependent on your velocity, which is dependent on your acceleration. It's self-referencing... I have no idea how to work this, just as I had no idea how to work pitch profile for ascent... So how do I work these?
P.S.: I'm trying to learn other orbital calculations by reading Fundamentals of Astrodynamics, which is not easy
Any advice there?
For a long time I've been wanting to be able to work calculations by hand for all the maneuvers conducted in orbiter, for it's own sake, but mostly to design realistic vehicles for Orbiter. I'm just getting into Calculus as a senior in high school, so lots of things are starting to open up. Recently I've discovered capital-sigma notation (aka summation notation), for instance (Well, on my own. So I don't really know it well...).
But one of the big mysteries I'm trying to get at right now is ascent trajectory, specifically the pitch program, how to mathematically solve to find the most efficient pitch program.
That's a long ways off... But for now I think I may have made one step, at least I know how to calculate gravitational drag! (I think)
I tested this in orbiter, and it was somewhat accurate, but for some reason there was an amount of error, that I'm thinking it may be because of the accuracy of the calculator I'm using (TI-84 Plus, it's trig isn't perfect)...
G = Cos(90/Ov * V) * GM/R^2
Cosine done in degrees.
G is the negative vertical acceleration due to gravity, in m/s^2.
Ov is orbital velocity
V is the vehicle's current horizontal velocity
And GM/R^2 is the equation for the force of gravity.
Is this correct? At least it seems to be, but my calculator and Orbiter aren't agreeing completely. To test it I took an ascent in the DG-S, and cut off the engines at different points to measure vertical acceleration due to gravity for different speeds. My error was usually in-between ~0.02 all the way up to ~0.1 m/s^2.
So to find the total gravitational drag, I simply plug this into summation notation (Which I don't know how to type on the forums :-/ ), with V as my changing variable, assuming a constant rate of horizontal acceleration...
Which is a problem. Since the rate of horizontal acceleration will depend on pitch and fuel used of the vehicle, it isn't anything near constant. So I still have no idea on how to find the most efficient pitch program.
And for the second part of my post:
I've been trying to find encounter velocity for orbital transfers. I need this for Delta-vee calculations of interplanetary and translunar vehicles. So what I did, is this:
Vapp = Sqrt(Vesc^2 * Vperi^2)
Vapp = Approach velocity
Vesc = Escape velocity at closest approach
Vperi = Velocity at closest approach (apogee of elliptical orbit around major body. In the case of TLI, Earth)
In TLI, I found I was about 200 m/s off, but it's hard to tell how accurate my math was, some of it may have been non-spherical gravity sources and course correction burns, seeing as TransX isn't perfect, but 200 m/s isn't really acceptable error for designing a Trans-Lunar Vehicle! And I don't even know if that equation actually works or not! Though it seems to, to some extant.
As exciting as the capital sigma (aka summation) notation was, I don't think it solves this problem of encounter velocity. Since as you fall towards a body, it's gravity will cause you to accelerate at a rate dependent on your distance, and your distance is dependent on your velocity, which is dependent on your acceleration. It's self-referencing... I have no idea how to work this, just as I had no idea how to work pitch profile for ascent... So how do I work these?
P.S.: I'm trying to learn other orbital calculations by reading Fundamentals of Astrodynamics, which is not easy
Any advice there?
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