Determining the oribital periods of the inner planets

Greated elongation etc. Source: wikimedia

Using a different approach, let's consider again two planets moving around the Sun in concentric circular orbits in the same plane.

Let the outer planet orbit the Sun at rate a (number of revolutions per time period).

Let the inner planet orbit the Sun at rate b (number of revolutions per time period).

At time t=0, let the planets be in the configuration shown such that viewed from the outer planet, the inner one is at greatest elongation (its greatest angular distance) from the Sun, in a specified direction. While in principle, we could have chosen inferior conjunction (when the inner planet is directly between the outer one and the Sun), in practice this is impossible to observe unless in the rare instance when the planet transits the Sun.

We can say that this configuration is repeated whenever:

(b - a).t(n) = n, where n is an integer > 0.

The time interval between the event at t(n) and the next, t(n+1), by definition is:

t(n+1)-t(n) = 1/(b - a)

This is independent of n. Let's call it 𝛥t.

Hence:

𝛥t = 1/(b - a)

So:

b/a = 1 + 1/(a.𝛥t)

Now, let's make this more concrete:

• the outer planet is Earth, so a is equal to 1 revolution per year
• the inner planet is Venus
• we'll choose the repeated event as the greatest elongation West, i.e. as observed in the morning sky - see table here.

From this table, we can see that an instance of greatest elongation West for Venus, observed from the Earth, occurred on the morning of March 22 2014 and then for a fifth time on almost the same day 8 years later on the morning of March 20 2022, so, 𝛥t is on average almost exactly 8/5 years.

Hence:

b/a = b ~ 1 + 5/8 = 13/8 (revolutions per year)

So, one orbit of Venus about the Sun takes 1/b, i.e. around 8/13 years or 224.8 days. The correct figure is 224.7 days, so we have achieved an accuracy of closer than 1 part in 1000.

Now consider Mercury in place of Venus. We can see there was an instance of greatest elongation West on March 14 2014 and then 22 instances later, another on March 6 2021. In this case:

b/a = b ~ 1 + 22/7 = 29/7 (revolutions per year)

Hence, one orbit of Mercury about the Sun takes 1/b, i.e. around 7/29 years or 88.2 days, which is close to the correct figure of 88.0 days, although with less accuracy than was the case for Venus. No doubt the relatively high inclination of its orbit to the ecliptic is responsible.

(We can adapt this method to the outer planets, with the inner planet now representing Earth and its rate of rotation, b, equal to 1. The rate of rotation of the outer planet, a, is given by:

a/b = 1 - 1/(b.𝛥t)

The time interval being measured is once more (the average) between one opposition and the next of the outer planet as seen from Earth.

)

Determining the orbital period of Jupiter to a high level of accuracy

Updated March 11

This sketch represents a view from above of the Earth in its orbit (green), around the Sun (bright orange in the original sketch!), and an outer planet, Jupiter say, (pink/red).

The orbits are circular and are in the same plane.

At time 1, for an observer on the Earth, Jupiter is said to be at opposition, when at its closest to the Earth.

At time 2, 6 months later, the Earth has completed half an orbit. Being much further away from the Sun than is the Earth, Jupiter's angular velocity is much lower and it has completed only a small part of its orbit in that time.

At time 3, Jupiter is at opposition again and since it has moved along in its orbit, the Earth also has to move along its orbit for an extra distance (solid green line), in a time interval represented by the delta symbol, 𝛥, in green, between the points marked 1 and 3. Jupiter has now completed 𝛥/365 of an orbit in 365 + 𝛥 days.

In 2026, opposition was January 10th and in 2027 it will be February 11th. In that time, Jupiter will have completed 32/365 of an orbit in 397 days. 

From this, its orbital period in years is therefore 397/32 or 12.4.        (Method 1)

The accepted period is 11.86, so what happened? Mainly it's because I've assumed the orbits are circular and they are not - they are elliptical. Now, I'm going to try to eliminate that error, by considering a longer period of time when the eccentricities of the orbits of Jupiter and Earth will have been smoothed out, somewhat.

Let's look ahead to 2038, when opposition is on January 14th and Jupiter will have completed 369/365 ((365+4)/365) of an orbit in 12 and 4/365 years. That means it will have completed a full orbit in 365/369 of those 12 and 4/365 years, i.e. its orbital period is:

((12x365) + 4)/369 or 11.88 years.        (Method 2)

That's very close to the accepted figure. I've not taken into account the exact times of day of the consecutive oppositions (available here), nor that the Earth takes 365 and 1/4 days to orbit the Sun. If I do that, then I make it 11.85 years.        (Method 3)

A similar analysis using method 2 above for Saturn over July 15th 1990 to July 21st 2020 gives an orbital period of 29.45 years (actual=29.46) and for Mars over March 4th 2012 to April 9th 2014 gives 1.91 years (actual=1.88). Interestingly, the deviation from the accepted period is greater in the latter case (closer to us, shorter time interval) and smaller in the former (further from us, bigger time interval).

The Coherer - the simplest detector of radio waves?

 

Here's a simple circuit consisting of a power supply, an LED and series resistor, in series with a coherer made of some scrunched up aluminium foil balls inside a plastic beaker in contact with each other and a couple of aluminium foil strips wired to complete the circuit. 

Except that the circuit is only completed once there is a nearby source of radio waves, which somehow creates a low resistance pathway for the electrons in the aluminium balls. In demonstrations, this is usually achieved with a piezo-electric gas igniter. However, in this video, made perhaps 18 months ago, I instead touch one of the foil strips with a short length of wire which acts as an aerial to pick up radio waves from, IIRC, the electric motor of an adjacent freezer. The LED (top left) then lights up. If I tap the side of the beaker sharply, the pathway is disrupted and the LED turns off again.

An even better calendar?

Here's a modified version of that new calendar. There are now fewer changes (6),  March and September having the same number of days as they do now:

January	31	30	-1	-1
February	28	29	+1	0
March	31	31	0	0
April	30	31	+1	+1
May	31	31	0	+1
June	30	31	+1	+2
July	31	31	0	+2
August	31	31	0	+2
September	30	30	0	+2
October	31	30	-1	+1
November	30	30	0	+1
December	31	30	-1	0

Let's see what difference this makes to the dates for equinoxes and solstices: 

• first, we'll fix the 2025 Autumn Equinox at Sep 22 (i.e. unchanged)
• the 2025 Winter Solstice is now on Dec 22, one day later than before, as there are now 90, not 91, days in total in the three months of September, October, November
• the 2026 Spring Equinox, almost exactly 89 days later, i.e. 3 full months, is on Mar 22
• the 2026 Summer Solstice, nearly 93 days later, again 3 full months, is on Jun 22
• the 2026 Autumn Equinox, again nearly 93 days later, and again 3 full months, is now on Sep 22.

Wow - all these events, at least for 2025/6, are now on the 22nd of the month, so this mk2 version is even better! I find that quite surprising given the somewhat lopsided look with 2 months with fewer than 31 days at the start of the year and 4 such months at the end.

An improved calendar?

Updated 28 Feb.

Following my last post, here is one suggestion for a new calendar, having a different number of days for 8 of the months in the year (highlighted). The current number of days per month is in the second column, the proposed number is in the third, the difference is in the fourth and the accumulated difference is in the fifth, which we'd expect to vary over the year, but to be 0 by the end, as indeed it is!

January	31	30	-1	-1
February	28	29	+1	0
March	31	30	-1	-1
April	30	31	+1	0
May	31	31	0	0
June	30	31	+1	+1
July	31	31	0	+1
August	31	31	0	+1
September	30	31	+1	+2
October	31	30	-1	+1
November	30	30	0	+1
December	31	30	-1	0

Let's see what difference this makes to the dates for equinoxes and solstices: 

• first, we'll fix the 2025 Autumn Equinox at Sep 22 (i.e. unchanged)
• the 2025 Winter Solstice is also unchanged on Dec 21, as there are still 91 days in total in the three months of September, October, November
• the 2026 Spring Equinox, almost exactly 89 days later is now on Mar 21, one day later in the calendar than now, as there are now 89 days in the three months December, January and February, as opposed to the actual 90
• the 2026 Summer Solstice, nearly 93 days later is also unchanged on Jun 21, as the number of days in total in March, April and May is still 92
• the 2026 Autumn Equinox, again nearly 93 days later is now on Sep 22, one day earlier in the calendar than now, as the number of days in total in June, July and August is now 93, rather than 92.

So, we can see that with the new equinox dates there would be less variation in these day numbers than before, i.e. on the 21st or 22nd of the month, as opposed to the 20th to 23rd (at least for the years quoted).

If there is a leap year, then we could give the extra day to February, as now, to make it a "proper" month with 30 days, so the number of days in any month would now only ever be 29, 30 or 31.

One final point - why not make December the shortest month, at 29 days and 30 in a leap year? Well, think of the potential for confusion, e.g. do we celebrate New Years Eve on the 29th or is there a 30th this year? January then? Well, it's so close to February, so let's stick with tradition.

The diminution of February

Updated 28 Feb.

My other half was pondering recently as to why February has so few days compared with the other months and with its immediate neighbours in particular. I responded that the Earth's orbit is not circular and during Winter, the Earth is a little closer to the Sun and moves more quickly, so something has to give at that time of year. Perhaps for some reason, it was decided that January and March were too important to give up any days, but February could take the hit instead and in a big way.

Anyway, I decided to look at the numbers. The dates and times (GMT) of recent and future equinoxes and solstices is as follows, with the last column being the time between the last event and current event (by my reckoning)*.

Autumn Equinox ‘25	22 Sep	18.19	-
Winter Solstice  ‘25	21 Dec	15.03	89 d, 20 h, 44 m
Spring Equinox  ‘26	20 Mar	14.46	88 d, 23 h, 43 m
Summer Solstice ‘26	21 Jun	08.24	92 d, 17 h, 38 m
Autumn Equinox ‘26	23 Sep	00.06	93 d, 15 h, 22 m

Those 'deltas' confirm the point I made and that it would make sense to have longer months in the Summer (by which I mean between the Spring and Autumn equinoxes), of generally 31 days, and shorter ones in Winter (by which I mean between the Autumn and Spring equinoxes), of generally 30 days, suitably adjusted to ensure 365 days per year and 366 in a leap year, of course. It should also mean less variation in the day number of the equinoxes and solstices.

(*Also by my reckoning, totalling those deltas makes about 365 and one quarter days, as we should expect).