English: Standard deviation of ecliptic longitudes of the five classical planets, from 2000 BC through AD 2201 (a period of 4200 years). The positions of Mercury, Venus, Mars, Jupiter, and Saturn were calculated for every tenth day using w:JPL Horizons On-Line Ephemeris System. The ecliptic longitude of the sun was then subtracted, so that the values for Mercury and Venus remain between -50° and +50°, and then the population standard deviation was taken (the root mean square of the deviation from the average). The graph was made using a smooth curve to connect the calculated points. The longitudes relative to the sun are smooth functions except when an outer planet passes opposition (180° from the sun), so when all the relative longitudes are smaller than say 90° a smooth curve drawn through evenly-spaced calculated values of the standard deviation will track well the true function. This graph shows the curve only when its value is less than 7°, where the smooth curve is quite accurate. Because of the scale of the x-axis, the places where the curve dips below 7° look like vertical lines. The data points are labeled with just the month and year (using the Julian Calendar before October 1582 and the Gregorian after that) because in any case they are not exactly at the date of the minimum of the dip. When the standard deviation reaches a "local" minimum all five planets are close together in the sky.

The largest longitude difference between two of the planets is not given by this method, but will be at least ${\displaystyle \sqrt{4\frac{1}{6}}}$ (≈2.04) times the standard deviation, which obtains when two planets are at one longitude and three at another. It is at most ${\displaystyle \sqrt{7\frac{1}{2}}}$ (≈2.74) times the standard deviation, which obtains when two planets are at one longitude, one is at the another, and the other two are at the longitude one-third of the way from the first to the second. (When the longitude difference is minimal, two of the planets have to be in true conjunction at one of the extremes of longitude.)

The grid lines are every 200 Julian years starting from January 1, 2000 BC.

The command lines for Horizons were: For the sun:

https://ssd.jpl.nasa.gov/api/horizons.api?format=text&COMMAND=Sun&START_TIME=%272000BC/1/1%2012:0%27&STOP_TIME=%271BC-Dec-07%27&STEP_SIZE=%2710%20days%27&QUANTITIES=31&ANG_FORMAT=DEG (for 2000 BC to 1 BC)

https://ssd.jpl.nasa.gov/api/horizons.api?format=text&COMMAND=Sun&START_TIME=%271BC/12/12%2012:0%27&STOP_TIME=%272201-Feb-07%27&STEP_SIZE=%2710%20days%27&QUANTITIES=31&ANG_FORMAT=DEG (for 1 BC to AD 2201)

For Mercury:

https://ssd.jpl.nasa.gov/api/horizons.api?format=text&COMMAND=1&START_TIME=%272000BC/1/1%2012:0%27&STOP_TIME=%271BC-Dec-07%27&STEP_SIZE=%2710%20days%27&QUANTITIES=31&ANG_FORMAT=DEG (for 2000 BC to 1 BC)

https://ssd.jpl.nasa.gov/api/horizons.api?format=text&COMMAND=1&START_TIME=%271BC/12/12%2012:0%27&STOP_TIME=%272201-Feb-07%27&STEP_SIZE=%2710%20days%27&QUANTITIES=31&ANG_FORMAT=DEG (for 1 BC to AD 2201)

For Venus:

https://ssd.jpl.nasa.gov/api/horizons.api?format=text&COMMAND=2&START_TIME=%272000BC/1/1%2012:0%27&STOP_TIME=%271BC-Dec-07%27&STEP_SIZE=%2710%20days%27&QUANTITIES=31&ANG_FORMAT=DEG (for 2000 BC to 1 BC)

https://ssd.jpl.nasa.gov/api/horizons.api?format=text&COMMAND=2&START_TIME=%271BC/12/12%2012:0%27&STOP_TIME=%272201-Feb-07%27&STEP_SIZE=%2710%20days%27&QUANTITIES=31&ANG_FORMAT=DEG (for 1 BC to AD 2201)

For Mars:

https://ssd.jpl.nasa.gov/api/horizons.api?format=text&COMMAND=4&START_TIME=%272000BC/1/1%2012:0%27&STOP_TIME=%271BC-Dec-07%27&STEP_SIZE=%2710%20days%27&QUANTITIES=31&ANG_FORMAT=DEG (for 2000 BC to 1 BC)

https://ssd.jpl.nasa.gov/api/horizons.api?format=text&COMMAND=4&START_TIME=%271BC/12/12%2012:0%27&STOP_TIME=%272201-Feb-07%27&STEP_SIZE=%2710%20days%27&QUANTITIES=31&ANG_FORMAT=DEG (for 1 BC to AD 2201)

For Jupiter:

https://ssd.jpl.nasa.gov/api/horizons.api?format=text&COMMAND=5&START_TIME=%272000BC/1/1%2012:0%27&STOP_TIME=%271BC-Dec-07%27&STEP_SIZE=%2710%20days%27&QUANTITIES=31&ANG_FORMAT=DEG (for 2000 BC to 1 BC)

https://ssd.jpl.nasa.gov/api/horizons.api?format=text&COMMAND=5&START_TIME=%271BC/12/12%2012:0%27&STOP_TIME=%272201-Feb-07%27&STEP_SIZE=%2710%20days%27&QUANTITIES=31&ANG_FORMAT=DEG (for 1 BC to AD 2201)

For Saturn:

https://ssd.jpl.nasa.gov/api/horizons.api?format=text&COMMAND=6&START_TIME=%272000BC/1/1%2012:0%27&STOP_TIME=%271BC-Dec-07%27&STEP_SIZE=%2710%20days%27&QUANTITIES=31&ANG_FORMAT=DEG (for 2000 BC to 1 BC)

https://ssd.jpl.nasa.gov/api/horizons.api?format=text&COMMAND=6&START_TIME=%271BC/12/12%2012:0%27&STOP_TIME=%272201-Feb-07%27&STEP_SIZE=%2710%20days%27&QUANTITIES=31&ANG_FORMAT=DEG (for 1 BC to AD 2201)

(Horizons didn't permit having as many dates as there would be doing the whole 4200 years in one go,)