Astronomy Answers: Seasons on the Planets

$$\def\|{&}$$

This page explains how you can calculate when the seasons begin on a planet.

According to the astronomical definition, a new season begins when the planetocentric ecliptic longitude $$λ_\text{Sun}$$ of the Sun is a multiple of 90°, like so:

Table 1: Seasons: Names

$${λ_\text{Sun}}$$ begins name code
North South
spring autumn ascending equinox I
90° summer winter northern solstice II
180° autumn spring descending equinox III
270° winter summer southern solstice IV

The "name" is an attempt from me to assign names that do not depend on the hemisphere where you are. The "code" is an even shorter label, useful to keep tables narrow.

For example: if the longitude is equal to 0°, then spring begins in the northern hemisphere and autumn in the southern hemisphere. We can call this moment the ascending equinox.

By rearranging some formulas from the calculation pages about the position of the Sun and the equation of Kepler, we find

\begin{align} ν \| = λ_\text{Sun} - Π + 180° \label{eq:ν} \\ \tan\left( \frac{E}{2} \right) \| = \sqrt{\frac{1 - e}{1 + e}} \tan\left( \frac{ν}{2} \right) \label{eq:E} \\ M \| = E - e \sin E \label{eq:M} \\ t \| = \frac{M - M_0}{M_1} \bmod \frac{360°}{M_1} \label{eq:t} \end{align}

where $$ν$$ is the true anomaly of the planet, $$Π$$ the planetocentric ecliptic longitude of the perihelion, $$e$$ the eccentricity of the orbit, $$M_0$$ the mean anomaly at a fixed moment, $$M_1$$ the rate of change of the mean anomaly, and $$t$$ the time since the fixed moment. In equation \ref{eq:M} the angles must be measured in radians. If you want to measure them in degrees instead, then replace $$e$$ by $$(e × 180°/π)$$ in equation \ref{eq:M}.

The reasonably fixed values are shown in the following table, with all angles measured in degrees and all time periods in Earth days. The fixed moment (for $$M_0$$) is 12:00 UTC on January 1st, 2000 = JD 2451545. I call those values "reasonably fixed" because they vary slowly with time, because the orbits of the planets themselves slowly change.

$${Π}$$$${e}$$$${M_0}$$$${M_1}$$$${360°/M_1}$$
Mercury 111.5943 0.20563 174.7948 4.09233445 87.969350
Venus 73.9519 0.00677 50.4161 1.60213034 224.70082
Earth 102.9372 0.01671 357.5291 0.98560028 365.25964
Mars 70.9812 0.09340 19.3730 0.52402068 686.99579
Jupiter 237.2074 0.04849 20.0202 0.08308529 4332.8970
Saturn 99.4571 0.05551 317.0207 0.03344414 10764.218
Uranus 5.4639 0.04630 141.0498 0.01172834 30694.881
Neptune 182.1957 0.00899 256.2250 0.00598103 60190.302
Pluto 4.5433 0.2490 14.882 0.00396 90909.091

If you enter these reasonably fixed values into the formulas, then you can calculate when the seasons begin.

As an example we'll calculate the dates of the southern solstices on Mars. With equation \ref{eq:ν} we find

$ν = 270° - 70.9812° + 180° = 379.0188° = 19.0188° \bmod 360°$

then with equation \ref{eq:E}

\begin{align*} \tan\left( \frac{E}{2} \right) \| = 0.91058 \tan(9.5094°) = 0.152532 \\ E \| = 17.3452° \end{align*}

then with equation \ref{eq:M}

$M = 17.3452° - 0.09340 × \frac{180°}{π} \sin 17.3452° = 15.7498°$

and then with equation \ref{eq:t}

$t = \frac{15.7498° - 19.3730°}{0.52402068°} \bmod \frac{360°}{0.52402068°} = −6.9142 \bmod 686.9958$

so on Mars the southern solstice (the beginning of winter in the northern hemisphere and summer in the southern hemisphere) happens −6.9142 days after (which is the same as 6.9142 days before) the fixed moment (at the beginning of the year 2000), and then again every 686.9958 days. The first time after the fixed moment was $$−6.9142 + 686.9958 = 680.0816$$ days after the fixed moment, which was on Julian Day number $$2451545 + 680.0816 = 2452225.0816$$ which corresponds to 11 november 2001.

If we do this for all seasons and for all planets, then we find for the first begin of the seasons after the beginning of the year 2000:

Table 2: Seasons: Dates

I II III IV
Mercury 2000-02-27T10:30 2000-03-22T23:56 2000-01-25T03:01 2000-02-12T07:50
Venus 2000-02-04T18:11 2000-04-01T12:48 2000-05-28T00:55 2000-07-22T14:47
Earth 2000-03-20T07:22 2000-06-21T01:41 2000-09-22T17:22 2000-12-21T13:40
Mars 2000-05-31T20:08 2000-12-16T10:41 2001-06-17T22:10 2001-11-11T13:57
Jupiter 2009-06-21T00:22 2000-04-29T02:06 2003-03-26T02:52 2006-06-16T00:43
Saturn 2009-08-08T15:28 2017-05-23T08:37 2025-05-13T19:20 2002-10-29T20:18
Uranus 2007-09-12T01:12 2030-01-24T00:46 2049-12-13T17:32 2069-08-14T03:31
Neptune 2046-07-08T15:13 2087-03-24T14:05 2128-11-17T14:05 2005-10-09T15:51
Pluto 2109-02-08T06:08 2192-12-30T04:47 2236-10-13T02:38 2029-08-13T09:07

These timestamps are written in an ISO 8601 format: first the date (year - month - day), then a T, and then the time (hour : minute) on a 24-hour clock. These times are listed in the UTC time zone. The corresponding Julian Day Numbers are:

Table 3: Seasons: JD

I II III IV
Mercury 2451601.9381 2451626.4976 2451568.6257 2451586.8266
Venus 2451579.2578 2451636.0335 2451692.5389 2451748.1162
Earth 2451623.8076 2451716.5703 2451810.2242 2451900.0701
Mars 2451696.3395 2451894.9456 2452078.4241 2452225.0816
Jupiter 2455003.5153 2451663.5878 2452724.6197 2453902.5299
Saturn 2455052.1448 2457896.8596 2460809.3058 2452577.3461
Uranus 2454355.5503 2462525.5324 2469789.2308 2476972.6471
Neptune 2468535.1346 2483404.0875 2498617.0868 2453653.1610
Pluto 2491394.7560 2522035.6994 2538027.6102 2462361.8801

We find the following lengths of the seasons (measured in Earth days):

Table 4: Seasons: Lengths

I II III IV
Mercury 24.5595 30.0974 18.2008 15.1116
Venus 56.7757 56.5054 55.5773 55.8424
Earth 92.7627 93.6540 89.8459 88.9971
Mars 198.606 183.478 146.658 158.254
Jupiter 992.969 1061.03 1177.91 1100.99
Saturn 2844.71 2912.45 2532.26 2474.80
Uranus 8169.98 7263.70 7183.42 8077.78
Neptune 14869. 15213. 15226.4 14882.
Pluto 30640.9 15991.9 15243.4 29032.9

languages: [en] [nl]

//aa.quae.nl/en/reken/seizoenen.html;
Last updated: 2017-12-28