Variable Gravity Acceleration.

[Andy44]

You don't need calculus to follow Scarecrow's post, it's all algebra.

If you want to propagate ephemeris, you should know the basics of calculus, but doing the integration in program code is not really calculus, it's an approximation of calculus.

I suggest trying Scarecrow's method, it's simpler and quicker. The main problem is not being able to do good math notation on this forum.

 

[perseus]

If you want to calculate the angle of reentry​

ANG =Atng(1/Tang(C)+(G.Mp/(sin(C).(R0.V^2-G.Mp))))

ANG= angle reetry
R0= perigee
V=speed reentry= sqr((1-cos(C)).Rr .G.Mp/(R0^2-R0.Rr.cos(c)))
Rr=radius reentry = 80Km + R Earth----->80000+35787000
C=angle tour of radius vector from perigee ]

Well ,here is the solution to calculate the speed and angle of reentry result of integration, I think that was what you needed​

 

[Miner1]

Scarecrow has the right idea. The energy and angular momentum are called integrals of motion (under the asumptions he mentioned, e.g. no drag, radial gravity, etc.). You can also mix and match the integrals of motion to get a new set of constants, e.g. semi-major axis, eccentricity, etc. By using the semi-major axis, a, and the eccentricity, e, you can simplify the solution a little bit. Here's how:

p = a*(1-e^2);
x = p/(e*Rei);
s = -sqrt(1-(x-1/e)^2);
fpa = atan(s/x);

where Rei is the radius at entry interface. You can also easily check to see if you are even going to reach entry interface. If x-1/e > 1, then your orbit won't reach Rei, i.e. your periapsis is too high. Notice also that the gravitational parameter, mu, doesn't enter in anywhere! So this calculation will work no matter where you happen to be orbiting.

I hope that helps a little.

 

[Scarecrow]

Still I don't get it...
Does anyone know calculus??

The problem with trying to integrate directly to a solution (the lots of calculus way) is that it requires more calculus than I know. I can do calculus in one variable just fine (which is how I got the potential energy of position formula, though I glossed over that bit), and a little multivariable calc, but I think your preferred method requires some sort of more advanced calculus, which I don't know yet.

I'm not sure exactly how to explain my way better, but I'll try to boil it down to just three formulas (each of which builds on the ones before):

velocity_later = square_root(velocity_now^2 - (2 * universal_gravitational_constant * mass_of_planet / distance_from_center_of_planet_now) + (2 * universal_gravitational_constant * mass_of_planet / distance_from_center_of_planet_later))

vertical_velocity_later = square_root(velocity_later^2 - ((distance_from_center_of_planet_now / distance_from_center_of_planet_later)^2 * (velocity_now^2 - vertical_velocity_now^2))

angle = asin(vertical_velocity_later / velocity_later)

"now" is right now, of course, and "later" is at some point in the future of your choosing (the entry interface). Since the only value you have to know ahead of time for "later" is the distance from the center of the planet, you choose this to be the altitude of the entry interface + the radius of the planet. You can look up the universal gravitational constant on Google.

Of course, these are just some formulas (which I think is what you want). If you want to know where they come from, see my previous post, and google whatever terms are unfamiliar.

I hope this helps.

By the way, I made a mistake in my previous post. I'm fixing it now, but if you looked at it before a message appears at the end of it that says it's fixed, don't trust what you find there. I corrected it for this post.

edit: It's interesting to note that the mass of the vessel cancelled out when I combined the formulas above. It makes sense that it should, since the answer shouldn't depend on the mass of the vessel, but it's interesting.

 

[Apokliptico]

Okay people, i've been studing the graphics of some kinematic funtions, and I think i've finnaly figured out the formula:
V = G0 * R^2 * T / (D^2+2*D*R+R^2) + G0 * T^2 * R / (2 * D^2 + 4 * R * D + R^2)

In which:
V = Velocity.
G0 = Gravity Acceleration at 0 m (9.78 m/s^2 for earth).
R = Planet mean radius.
T = Time to hit the atmosphere (could be the PeT).
D = Distance at the time of sampling.

Please test it and see if it's OK.