Turn a sinusoid into a elliptic orbit

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In summary: 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)wherevr = (vx^2+vy^2+vz^2)^0.5Neither one.
  • #1
Philosophaie
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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)
 
Last edited:
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  • #2
Philosophaie said:
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.
 
  • #3
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
 
  • #4
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.
 
  • #5
Philosophaie said:
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.
Philosophaie said:
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?
 
  • #6
Philosophaie said:
I am not sure they are correct.
They are not correct.
 
  • #7
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?
 
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  • #8
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.
 

1. How do you turn a sinusoid into an elliptic orbit?

In order to turn a sinusoid into an elliptic orbit, you will need to apply a transformation to the sinusoidal function. This transformation involves changing the amplitude, period, and phase of the sinusoid to match the characteristics of an elliptic orbit.

2. What is the difference between a sinusoid and an elliptic orbit?

A sinusoid is a mathematical function that describes a repetitive oscillation, while an elliptic orbit is the path that a celestial body follows around another body in space. A sinusoid can be transformed into an elliptic orbit to model the motion of a celestial body.

3. Can any sinusoid be turned into an elliptic orbit?

Yes, any sinusoidal function can be transformed into an elliptic orbit by adjusting its parameters. However, the resulting orbit may not accurately represent the motion of an actual celestial body.

4. What are the benefits of turning a sinusoid into an elliptic orbit?

Transforming a sinusoidal function into an elliptic orbit allows us to model the motion of a celestial body more accurately. This can help us understand the behavior of celestial bodies and make predictions about their future positions and movements.

5. How is a sinusoid transformed into an elliptic orbit used in scientific research?

In scientific research, transforming a sinusoidal function into an elliptic orbit is used to study the motion of celestial bodies and make predictions about their future positions. This information is crucial for space exploration and understanding the universe.

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