When do Mars, Venus and Earth Align?

Question

I found that all planets in the solar system align on 6 May 2492. How can I find when Mars, Venus and Earth align? Perhaps it is sooner.

Answer 1

JPL's HORIZONS Web-Interface gives you access to the positions of planets (among many other things) for a large time period for Earth and Venus, but only until the year 2500 for Mars.

Update:

I found 2400-Nov-28 to be the closest alignment for Mars, Earth, and Venus (from now until 2500, 10 day spacing):

Also of interest, the RMS values I used to find the closest alignment:

Answer 2

I found that all planets in the solar system align on 6 May 2492.

I found that at least the first four planets in the solar system DO NOT align on 6 May 2492, but they are at least on the same side of the Sun.

From Horizons:

Object     date/time UTC           X              Y              Z
-------  -----------------   ------------   -------------   -------------
Sun      2492-May-06 00:00,  2.918640E+05,  1.1212820E+06, -1.3467326E+04
Mercury  2492-May-06 00:00,  3.369897E+07, -5.4471967E+07, -7.6141609E+06
Venus    2492-May-06 00:00,  3.058573E+07, -1.0337403E+08, -3.3188750E+06
Earth    2492-May-06 00:00, -1.153599E+08, -9.5367510E+07,  1.0807226E+05
Mars     2492-May-06 00:00, -4.014225E+07, -2.1701158E+08, -3.6649254E+06

Update:

I used Python and Skyfield and DE405 to get all the planets. There is no such alignment.

from skyfield.api import Loader
import numpy as np
import matplotlib.pyplot as plt

from skyfield.api import load
loaddata = Loader('~/Documents/fishing/SkyData')  # avoids multiple copies of large files

ts = loaddata.timescale() # include builtin=True if you want to use older files (you may miss some leap-seconds)
eph = loaddata('de405.bsp')

things = ('sun', 'mercury', 'venus', 'earth', 'mars barycenter',
           'jupiter barycenter', 'saturn barycenter',
           'uranus barycenter', 'neptune barycenter', 'pluto barycenter')

times = ts.utc(2049, 5, 6)

objects = [eph[x] for x in things]

positions = np.array([ob.at(times).position.km for ob in objects])
plt.figure()
for x, y, z in positions:
    plt.plot(x, y, 'o')
plt.gca().set_aspect('equal')
plt.title('2492-May-06 00:00 Sun + planets + pluto')
plt.xlabel('X ecliptic (km)')
plt.ylabel('Y ecliptic (km)')
plt.show()

Answer 3

This seems to me like it has a relatively easy exact answer. I don't have my orbital mechanics book, but this is a clock. I'm working with periods that only to 5 digits, but in days (from https://space-facts.com/orbital-periods-planets/) I get the following orbital periods:

venus 224.70 days

earth 365.26 days

mars 686.98 days

The ratio of mars and earth (the slowest moving) determines the "tick" of the clock. Every mars year, the earth will have made nearly 2 revolutions (1.8808ish, actually). Earths orbital angular velocity is faster than that of Mars, but what we want is the point at which Earth has made more than one revolution, but less than two revolutions. Earth's angular velocity is 1.88088 times that of Mars, so if we start aligned, and choose this as a zero angle, then Earths angle (in revolutions) at any time t (in Mars years) is

$\theta_e = 1.88088t$

Where Mar's angle is given by

$\theta_m = 1.0t$

and earth will be back at the starting point when t=1/1.88088=.53168, just bit more than 1/2 mars year. During this first earth year, Mars is moving more slowly than earth, and they have not aligned again. Resetting the epoch, we have a new expression for Earth's angle at the start of the new earth year:

$\theta_e = 1.8808(t-.53168)$

Now, we can find the solution for the time when Earth and Mars align again. Setting \theta_e and \theta_m equal, we now have them aligning when

$t = 1.8808(t-.53168) \quad\mapsto\quad t = 1.8808*0.53168/0.8808 = 1.135313$

In other words, earth/mars alignment will occur every 1.135313 Mars years. (The number of earth revolutions, is 2.135313 from the first equation, so I haven't slipped anything up yet).

Now, this period is the tick of the clock for the 3 body system. And I'm pretty sure 5 digits are insufficient accuracy to do the last bit, but find a point at which earth mars align exactly, then every 1.135313 Mars years, or 2.135313 Earth years, Venus will advance by a fixed phase (removing the integer portion of revolutions). I get this as

1.135313060853769300*224.70/686.98 = .37134 (Revolutions)

Now, (with better orbital parameters) and the initial angular position of Venus relative to the line containing Earth and Mars, it's a simple matter to print out the relative phase of Venus at every Earth/Mars alignment, and pick your criterion for "alignment" of the 3 body system.

Final bit, the easy exact answer is never, of course. Or so I'd surmise.

Answer 4

This isn't an exact alignment, but tomorrow night (July 12, 2021), Venus and Mars only be 0.5 degrees apart in the night sky!