How do I plot a planet's orbit as a function of time on an already plotted ellipse?

Question

I am trying to create a program for my course where the user inputs a time after 1/1/16 and the program will run through the planet's cycle as many times as it needs to and then plot an image of the orbit with the planet in the correct location. My program doesn't need to be too complex so i've ignored shifts over long periods etc. I simply need to know a way (possibly using integration) to map a point on an ellipse and have the varying speed of the planet taken into account. I know that the area mapped out by two points and one focal point remains the same as long as the two points are sampled at the same time interval so thats where my thought about integrating the lines equation might be useful but Im kind of out of my depth.

This is my code so far, thanks for any help you can give:

import matplotlib.pyplot as plt
from matplotlib.patches import Ellipse
#Set axes aspect to equal as orbits are almost circular; hence square is needed
ax = plt.figure(0).add_subplot(111, aspect='equal')

#Setting the title, axis labels, axis values and introducing a grid underlay
#Variable used so title can indicate user inputed date
plt.title('Inner Planetary Orbits at[user input date]')
plt.ylabel('x10^6 km')
plt.xlabel('x10^6 km')
ax.set_xlim(-300, 300)
ax.set_ylim(-300, 300)
plt.grid()

#Creating the point to represent the sun at the origin (not to scale), 
ax.scatter(0,0,s=200,color='y')
plt.annotate('Sun', xy=(0,-30))

#Implementing ellipse equations to generate the values needed to plot an ellipse
#Using only the planet's min (m) and max (M) distances from the sun
#Equations return '2a' (the ellipses width) and '2b' (the ellipses height)
def OrbitLength(M, m):
    a=(M+m)/2
    c=a-m
    e=c/a
    b=a*(1-e**2)**0.5
    print(a)
    print(b)
    return 2*a, 2*b

#This function uses the returned 2a and 2b for the ellipse function's variables
#Also generating the orbit offset (putting the sun at a focal point) using M and m
def PlanetOrbit(Name, M, m):
    w, h = OrbitLength(M, m)
    Xoffset= ((M+m)/2)-m
    Name = Ellipse(xy=((Xoffset),0), width=w, height=h, angle=0, linewidth=1, fill=False)
    ax.add_artist(Name)

#These are the arguments taken from hyperphysics.phy-astr.gsu.edu/hbase/solar/soldata2.html
#They are the planet names, max and min distances, and their longitudinal angle
#Also included is Halley's Comet, used to show different scale  and eccentricity
PlanetOrbit('Mercury', 69.8, 46.0)
PlanetOrbit('Venus', 108.9, 107.5)
PlanetOrbit('Earth', 152.1, 147.1)
PlanetOrbit('Mars', 249.1, 206.7)
PlanetOrbit("Halley's Comet",45900,88)

Tamas Hegedus · Accepted Answer · 2021-12-19T23:24:44.083

Found something similar at this gamedev question. Someone there linked this wikipedia article, which describes a method step by step how to calculate the position (in heliocentric polar coordinates) as a function of time. The equations contain Newton's gravitational constant and the mass of the Sun of course. Some of them are only solveable by numeric methods.

Implementation of the described method:

from math import *

EPSILON = 1e-12
def solve_bisection(fn, xmin,xmax,epsilon = EPSILON):
  while True:
      xmid = (xmin + xmax) * 0.5
      if (xmax-xmin < epsilon):
        return xmid
      fn_mid = fn(xmid)
      fn_min = fn(xmin)
      if fn_min*fn_mid < 0:
          xmax = xmid
      else:
          xmin = xmid
    
'''
Found something similar at this gamedev question:
https://gamedev.stackexchange.com/questions/11116/kepler-orbit-get-position-on-the-orbit-over-time?newreg=e895c2a71651407d8e18915c38024d50

Equations taken from:
https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#Position_as_a_function_of_time
'''
def SolveOrbit(rmax, rmin, t):
  # calculation precision
  epsilon = EPSILON
  # mass of the sun [kg]
  Msun = 1.9891e30
  # Newton's gravitational constant [N*m**2/kg**2]
  G = 6.6740831e-11
  # standard gravitational parameter
  mu = G*Msun
  # eccentricity
  eps = (rmax - rmin) / (rmax + rmin)
  # semi-latus rectum
  p = rmin * (1 + eps)
  # semi/half major axis
  a = p / (1 - eps**2)
  # period
  P = sqrt(a**3 / mu)
  # mean anomaly
  M = (t / P) % (2*pi)
  # eccentric anomaly
  def fn_E(E):
    return M - (E-eps*sin(E))
  E = solve_bisection(fn_E, 0, 2*pi)
  # true anomaly
  # TODO: what if E == pi?
  theta = 2*atan(sqrt((((1+eps)*tan(E/2)**2)/(1-eps))))
  # if we are at the second half of the orbit
  if (E > pi):
    theta = 2*pi - theta
  # heliocentric distance
  r = a * (1 - eps * cos(E))
  return theta, r

def DrawPlanet(name, rmax, rmin, t):
  SCALE = 1e9
  theta, r = SolveOrbit(rmax * SCALE, rmin * SCALE, t)
  x = -r * cos(theta) / SCALE
  y = r * sin(theta) / SCALE
  planet = Circle((x, y), 8)
  ax.add_artist(planet)

So the whole program looks like something like this:

import matplotlib.pyplot as plt
from matplotlib.patches import Ellipse, Circle

#Set axes aspect to equal as orbits are almost circular; hence square is needed
ax = plt.figure(0).add_subplot(111, aspect='equal')

#Setting the title, axis labels, axis values and introducing a grid underlay
#Variable used so title can indicate user inputed date
plt.title('Inner Planetary Orbits at[user input date]')
plt.ylabel('x10^6 km')
plt.xlabel('x10^6 km')
ax.set_xlim(-300, 300)
ax.set_ylim(-300, 300)
plt.grid()

#Creating the point to represent the sun at the origin (not to scale), 
ax.scatter(0,0,s=200,color='y')
plt.annotate('Sun', xy=(0,-30))

#Implementing ellipse equations to generate the values needed to plot an ellipse
#Using only the planet's min (m) and max (M) distances from the sun
#Equations return '2a' (the ellipses width) and '2b' (the ellipses height)
def OrbitLength(M, m):
    a=(M+m)/2
    c=a-m
    e=c/a
    b=a*(1-e**2)**0.5
    print(a)
    print(b)
    return 2*a, 2*b

#This function uses the returned 2a and 2b for the ellipse function's variables
#Also generating the orbit offset (putting the sun at a focal point) using M and m
def PlanetOrbit(Name, M, m):
    w, h = OrbitLength(M, m)
    Xoffset= ((M+m)/2)-m
    Name = Ellipse(xy=((Xoffset),0), width=w, height=h, angle=0, linewidth=1, fill=False)
    ax.add_artist(Name)

    
    
from math import *

EPSILON = 1e-12
def solve_bisection(fn, xmin,xmax,epsilon = EPSILON):
  while True:
      xmid = (xmin + xmax) * 0.5
      if (xmax-xmin < epsilon):
        return xmid
      fn_mid = fn(xmid)
      fn_min = fn(xmin)
      if fn_min*fn_mid < 0:
          xmax = xmid
      else:
          xmin = xmid
    
'''
Found something similar at this gamedev question:
https://gamedev.stackexchange.com/questions/11116/kepler-orbit-get-position-on-the-orbit-over-time?newreg=e895c2a71651407d8e18915c38024d50

Equations taken from:
https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#Position_as_a_function_of_time
'''
def SolveOrbit(rmax, rmin, t):
  # calculation precision
  epsilon = EPSILON
  # mass of the sun [kg]
  Msun = 1.9891e30
  # Newton's gravitational constant [N*m**2/kg**2]
  G = 6.6740831e-11
  # standard gravitational parameter
  mu = G*Msun
  # eccentricity
  eps = (rmax - rmin) / (rmax + rmin)
  # semi-latus rectum
  p = rmin * (1 + eps)
  # semi/half major axis
  a = p / (1 - eps**2)
  # period
  P = sqrt(a**3 / mu)
  # mean anomaly
  M = (t / P) % (2*pi)
  # eccentric anomaly
  def fn_E(E):
    return M - (E-eps*sin(E))
  E = solve_bisection(fn_E, 0, 2*pi)
  # true anomaly
  # TODO: what if E == pi?
  theta = 2*atan(sqrt((((1+eps)*tan(E/2)**2)/(1-eps))))
  # if we are at the second half of the orbit
  if (E > pi):
    theta = 2*pi - theta
  # heliocentric distance
  r = a * (1 - eps * cos(E))
  return theta, r

def DrawPlanet(name, rmax, rmin, t):
  SCALE = 1e9
  theta, r = SolveOrbit(rmax * SCALE, rmin * SCALE, t)
  x = -r * cos(theta) / SCALE
  y = r * sin(theta) / SCALE
  planet = Circle((x, y), 8)
  ax.add_artist(planet)
    
#These are the arguments taken from hyperphysics.phy-astr.gsu.edu/hbase/solar/soldata2.html
#They are the planet names, max and min distances, and their longitudinal angle
#Also included is Halley's Comet, used to show different scale  and eccentricity
PlanetOrbit('Mercury', 69.8, 46.0)
PlanetOrbit('Venus', 108.9, 107.5)
PlanetOrbit('Earth', 152.1, 147.1)
PlanetOrbit('Mars', 249.1, 206.7)
PlanetOrbit("Halley's Comet",45900,88)
for i in range(0, 52):
  DrawPlanet('Earth', 152.1, 147.1, i/52 * 365.25 *60*60*24)
for i in range(-2, 3):
  DrawPlanet("Halley's Comet",45900,88, 7*i *60*60*24)
print(60*60*24*365)
  
plt.show()

Edit: Thanks to @kdubs, the snippet is now using the closed form solution for theta.

@LiamWheen If you find my answer helpful you can even accept it :) — Tamas Hegedus, Jan 02 '16 at 17:52
I found you can calculate theta like this : `theta = 2*atan(sqrt((((1+eps)*tan(E/2)**2)/(1-eps))))` removing the need for calling solve_bisector for that part of the code — kdubs, Dec 16 '21 at 04:34

How do I plot a planet's orbit as a function of time on an already plotted ellipse?

1 Answers1