Turn a sinusoid into a elliptic orbit


by Philosophaie
Tags: elliptic, orbit, sinusoid, turn
Philosophaie
Philosophaie is offline
#1
Sep20-13, 11:38 AM
P: 365
I am trying to get my head around these equations. I am not sure they are correct. My logic is an orbit exists at a starting point (x0,y0,z0) with a starting velocity at time zero (vx0,vy0,vz0) changing with time (dx/dt, dy/dt, dz/dt). The gravity is (d^2x/dt^2, d^2y/dt^2, d^2z/dt^2). How do you turn a sinusoid into a elliptic orbit?

x = (-1 / 2 * G * M / r ^ 2 * Cos(h) * Cos(p)) * t ^ 2 + vx0 * t + x0
y = (-1 / 2 * G * M / r ^ 2 * Sin(h) * Cos(p)) * t ^ 2 + vy0 * t + y0
z = (-1 / 2 * G * M / r ^ 2 * Sin(p)) * t ^ 2 + vz0 * t + z0

where
r=(x^2+y^2+z^2)^0.5
h=atan(y/x)
p=acos(z/r)
Phys.Org News Partner Astronomy news on Phys.org
Quest for extraterrestrial life not over, experts say
Continents may be a key feature of Super-Earths
Astronomers discover first self-lensing binary star system
Simon Bridge
Simon Bridge is offline
#2
Sep20-13, 07:38 PM
Homework
Sci Advisor
HW Helper
Thanks ∞
PF Gold
Simon Bridge's Avatar
P: 10,990
Quote Quote by Philosophaie View Post
I am trying to get my head around these equations. I am not sure they are correct. My logic is an orbit exists at a starting point (x0,y0,z0) with a starting velocity at time zero (vx0,vy0,vz0) changing with time (dx/dt, dy/dt, dz/dt). The gravity is (d^2x/dt^2, d^2y/dt^2, d^2z/dt^2). How do you turn a sinusoid into a elliptic orbit?
The sinusoids parameterise the ellipse.
You know how this works if the plane of the orbit is the x-y plane right?

However, I think you have been too general in your setup.
Gravity is usually a central force - always directed to some point - with a magnitude that depends on the distance to that center. You write that down and apply Newton's Laws. There are several possible shapes - the stable ellipse is usually quite tricky to hit on by trail and error.
Philosophaie
Philosophaie is offline
#3
Sep22-13, 02:11 AM
P: 365
Which is the correct postulation in Newtonian 2 Body solution:

h=atan(y/x)
p=acos(z/r)

or

h=atan(vy/vx)
p=acos(vz/vr)

where

vr = (vx^2+vy^2+vz^2)^0.5

Simon Bridge
Simon Bridge is offline
#4
Sep22-13, 03:28 AM
Homework
Sci Advisor
HW Helper
Thanks ∞
PF Gold
Simon Bridge's Avatar
P: 10,990

Turn a sinusoid into a elliptic orbit


Depends what you want h and p to represent.
The former are the angles to the position and the second to the velocity.
They do not appear to represent any kind of postulates, and are not specific to the two-body problem.
D H
D H is offline
#5
Sep22-13, 06:37 AM
Mentor
P: 14,454
Quote Quote by Philosophaie View Post
I am trying to get my head around these equations. I am not sure they are correct. My logic is an orbit exists at a starting point (x0,y0,z0) with a starting velocity at time zero (vx0,vy0,vz0) changing with time (dx/dt, dy/dt, dz/dt). The gravity is (d^2x/dt^2, d^2y/dt^2, d^2z/dt^2). How do you turn a sinusoid into a elliptic orbit?

x = (-1 / 2 * G * M / r ^ 2 * Cos(h) * Cos(p)) * t ^ 2 + vx0 * t + x0
y = (-1 / 2 * G * M / r ^ 2 * Sin(h) * Cos(p)) * t ^ 2 + vy0 * t + y0
z = (-1 / 2 * G * M / r ^ 2 * Sin(p)) * t ^ 2 + vz0 * t + z0

where
r=(x^2+y^2+z^2)^0.5
h=atan(y/x)
p=acos(z/r)
This is incorrect. ##\vec x(t) = \frac 1 2 \vec a\,t^2 + \vec v_0\,t + \vec x_0## is only valid for constant acceleration. The equation you used does not apply to an orbiting body because the acceleration of an orbiting body is not constant.


Quote Quote by Philosophaie View Post
Which is the correct postulation in Newtonian 2 Body solution:

h=atan(y/x)
p=acos(z/r)

or

h=atan(vy/vx)
p=acos(vz/vr)

where

vr = (vx^2+vy^2+vz^2)^0.5
Neither one.

The correct solution was found by Kepler. Why do you persist in avoiding that solution?
DaleSpam
DaleSpam is offline
#6
Sep22-13, 06:46 AM
Mentor
P: 16,477
Quote Quote by Philosophaie View Post
I am not sure they are correct.
They are not correct.
Philosophaie
Philosophaie is offline
#7
Sep23-13, 04:56 AM
P: 365
Then how do you find the equations for x,y,z, xdot, ydot, zdot, rdot thetadot, phidot, xdoubledot, ydoubledot, zdoubledot, rdoubledot, thetadoubledot, and phidoubledot with non-uniform acceleration?
Simon Bridge
Simon Bridge is offline
#8
Sep23-13, 07:44 PM
Homework
Sci Advisor
HW Helper
Thanks ∞
PF Gold
Simon Bridge's Avatar
P: 10,990
Start with a free-body diagram and apply Newton's Laws - applying boundary conditions.

From post #2:
Gravity is usually a central force - always directed to some point - with a magnitude that depends on the distance to that center. You write that down and apply Newton's Laws. There are several possible shapes - the stable ellipse is usually quite tricky to hit on by trail and error.


Register to reply

Related Discussions
Elliptic orbit of a Satellite Introductory Physics Homework 20
elliptic orbit of earth round the sun General Astronomy 1
Elliptic orbit proofs Advanced Physics Homework 0
Circular-elliptic orbit Advanced Physics Homework 8
Elliptic orbit from empty focus Advanced Physics Homework 0