Jupiter vs. the Sun

To compare the Jovian Kepler constant with the Solar one, we must convert it to the same units (AUs cubed over years squared):

• we determine the diameter of Jupiter is ~ 0.001 AUs by estimating* its angular diameter is about 3/4 of one minute of arc at opposition when we know the planet is 4.1 AUs away (its orbital radius minus that of the Earth's)
• we cube this: ~ 10⁻⁹ and multiply our Jovian Kepler constant by it
• we determine the period of moon I (Io) is ~1.8/365.25 or 0.0049 years
• we square this: ~ 2.4 x 10⁻⁵ and divide the above result by it

(* If there were two stars of known angular separation visible in the eyepiece of the telescope at the same time as Jupiter, their presence could provide a useful yardstick against which to make a reasonably accurate estimate).  

Hence, the average Kepler constant for the Jovian system is ~  0.001, i.e. 1/1000th of the value (1) for the Solar System. This means that for two different objects orbiting at the same distance from the Sun and from Jupiter, in the former case the object's speed is 10√10 times that of the latter. (Similarly, if they have the same orbital period, then the object orbiting the Sun does so at 10 times the distance of that orbiting Jupiter). 

We conclude that the Sun's gravitational pull is much stronger than Jupiter's and that the ratio of their Kepler constants is a measure of this.

Given its angular diameter at opposition is as stated earlier, then we conclude that at the distance of the Sun (1 AU), it would be 4.1 times larger, i.e. ~ 0.05°, compared with the Sun's  ~ 0.5°. That is, Jupiter's diameter is about 10% that of the Sun and its volume is about one-thousandth, i.e. about the same as the ratio of its Kepler constant to the Sun's.

Here then is a first inkling about what may be behind the strength of a body's gravitational pull. Volume seems an unlikely candidate as we shouldn't expect a diffuse cloud of gas to have the same effect as a solid object of the same size. Perhaps the gravitational pull is simply proportional to its mass then? If so, then Jupiter's density is about the same as the Sun's. And yet, what different bodies they seem to be!

The Jovian Keplerian system

Updated 11 May 26

The discovery of Jupiter's moons through the newly-invented telescope back in the early 1600s must have caused great excitement at the time and a wish to investigate further to see if the Jovian system follows Kepler's laws.

However, imagine trying to track these moons individually over many days and weeks in an attempt to measure the orbital radius and orbital period for each!

The orbital radius could be estimated crudely (being imprecise and prone to human error) using the diameter of the planet as the base unit, e.g. the innermost moon being found to orbit at around 3 Jovian diameters from the centre of the planet.

I've tried this myself on just a few occasions with binoculars and it was difficult! Doing this to an acceptable level of accuracy and precision in the early 17th century would have demanded the best optics of the time, a solid mount, good skies over many weeks and an experienced and patient observer. And then no doubt repeating the exercise again and again. 

The orbital period could be measured with more precision by timing the intervals between successive identical locations in a moon's orbit (e.g. maximum elongation), although we should bear in mind that this was before the invention of the pendulum clock.

(Would we have to allow for the movement of the Earth introducing errors due to parallax?)

See the table at the bottom of the page for an investigation. (The table had to be imported and in this case, if not placed there, messes up the font of the text following it!). The moons are numbered according to their proximity to Jupiter.

It turns out that the orbital periods of moons I, II and III are in the ratio 1:2:4, so I'm using the orbital period of moon I as the unit of time here.

The average estimate of the Jovian Kepler 'constant' (last column) is 26, but with a large range of values, no doubt due to the lack of precision in measuring orbital radii in whole Jovian diameters, which are then cubed. (In practice, there would be even greater variation with this method, because, unlike the case here, estimates would likely be inaccurate, as well as imprecise). However, these values are of a similar order of magnitude and a more precise and more accurate method should yield results that are more consistent*. Therefore, we tentatively assert that these moons do indeed obey Kepler's third law.

(* If instead we were to estimate accurately using the radius of Jupiter as our measuring unit, then we should find that the Jovian Kepler constant has an average of 202 with a proportionately smaller range of 182 to 216. However, I expect this advantage would have been easily removed by increased human error, in estimating, say, that Callisto orbits at 26 units from Jupiter! I imagine that the later invention of the reticule would have been revolutionary for making accurate and precise measurements). 

Moon	Orbital radius (r) in Jovian diameters	r^3	Period (t) as a multiple of that of moon I	t^2	r^3/t^2
I. Io	3	27	1	1	27
II. Europa	5	125	2	4	31
III.Ganymede	7	343	4	16	21
IV. Callisto	13	2197	9.4	89	25

Start of Summer

Updated 1 May 26

The Spring equinox this year was on the 20th of March. With the Summer solstice on the 21st of June, I estimate the mid-point - and surely the proper start of Summer -  will be about the 5th of May. However, the first of May seems a good enough time to celebrate it.

Traditionally, May Day is associated with fertility, so we have the symbolism of the May Pole and the May Queen in many European countries. In Beltane, there is a tradition of lighting bonfires, no doubt to symbolically to drive away the dark and the evil spirits (rather than to defiantly face them as done at Halloween).

In Germany (and elsewhere in Europe), the celebration has been co-opted by the Church, to celebrate Saint Walpurga, an Anglo-Saxon nun tasked with converting the Germanic tribes in the eighth century. There are bonfires the night before (Walpurgisnacht in Germany), indicating a common root with Beltane. Given Walpurga is known for having fought against witchcraft, she was an inspired choice for re-branding the event!

As noted in an earlier post, I've read that having four seasons in the year is a fairly recent innovation and that traditionally there were just Winter and Summer. (Ireland seems to be the exception, but some scholars have said that the first references to Imbolc were by Christian writers, casting doubt on an ancient four seasons tradition). However, if that is so, why celebrate or note the mid-points between solstices and equinoxes?

In addition, it seems there was a Germanic and Nordic custom of Summer starting with May Day and ending with Halloween. Perhaps this was nothing to do with seasonal mid-points, but because of the time it takes for earth and sea to heat up and to cool down again, although that seems to be at odds with the fire festivals, which suggest they were about daylight. Either way, mid-Summer and mid-Winter celebrations would still have been held on the solstices and not early August (Lammsas) and early February (Imbolc).

Perhaps, in the Germanic tradition, Summer and Autumn (albeit the latter un-named) were combined to form one large season with Winter and Spring (also un-named) the other? 

Measuring the elongation of Venus with a watch?

Updated 3 May 26

As an alternative, I thought instead about just using the timings of the settings of the Sun and of Venus (I'm not a morning person!). Thinking that no doubt someone has already investigated this, after some digging, I came across this site with a method for such a measurement. However, there's no derivation of the method and it seems to assume that both will set at the same azimuth, which doesn't seem right. Anyway, I thought I'd check it with Stellarium and chose the greatest elongation East for Venus of 15 August 2026:

• Sun and Venus are about 45° apart, as we'd expect
• the Sun's declination (angular distance above the celestial equator) is around 14°
• that of Venus is around -5° (i.e. it's below the celestial equator), so very different from that of the Sun
• the Sun straddles the horizon, i.e. is half set at around 20.20 (BST?) at azimuth 294°
• Venus sets about 80 minutes later, at azimuth 261°, i.e. there's a whopping 33° difference!

If we use the formula, it says for a 45° separation, Venus sets 3 hours, not 1 hour, after the Sun, so the formula is indeed wrong - the Sun and Venus will set at the same point on the horizon only if they are at the same declination, which is not the case here nor in general. As a comparison, the star Denebola *is* at about the same declination as the Sun at this time, with an angular separation from it of 31°. The formula gives about 2 hours between the Sun and Denebola setting and Stellarium confirms that.

In addition to the time between the setting of the Sun and of Venus, then we'd need the difference in azimuths at setting or the declinations - and a new formula.

Verdict: What's the point?

Measuring the elongation of Venus

Updated 30 Apr 26

Given Mercury's proximity to the Sun, let's take it that measuring elongation will always be a challenge, so let's limit our investigation to Venus for now.

Measuring the elongation of Venus when it's distant from the Sun may seem easy enough in principle, if you have a clear view of the horizon -  just measure the direction on the horizon (azimuth) of the centre of the setting Sun and at the same time, record the azimuth of Venus and its altitude above the horizon and then determine the angular difference. However, given that the Sun will be bright and Venus much less so, perhaps it's going to be difficult in practice to observe both at the same time?

Wait then until the Sun has almost fully set, with just a remnant showing on the horizon, and Venus may be more obvious. Take measurements as before, but note the Sun's altitude is now one solar diameter (~ 0.5°) below the horizon, which we should probably take into account.

However,  refraction by the Earth's atmosphere will also make a difference to the measurements of azimuth of both Venus and of the Sun. At the moment we don't know how to quantify this effect, but hope that it will be similar in both. Let's not worry too much about this and consider our measurement an estimate only. One or two degrees in measuring greatest elongation is unlikely to make a massive difference to our estimate of the radius of the orbit of Venus, although it will of course have more significance for estimating that of Mercury.

Determining the orbital radius for Saturn

Updated 5 May 26

To complete the orbital radii of the classical planets, let's now find Saturn's:

rₛ ~ (dₑ + sin θ)/ ((dₑ/pₛ) + sin θ)

where pₛ is its orbital period, for which we'll give the value of 29.45 that we found earlier.

This time, let's double Δt to 4 days, because we know θ may be too small to measure otherwise. As a result, distance dₑ will now be 8π/365.25 AUs. However, as this is an approximation (chord vs. arc of a circle), we will have increased the error compared with a time of 2 days.

Considering the opposition of Saturn on October 4 2026 and using Stellarium to determine its position 48 hours before and 48 hours after, and this tool, θ, the angular distance Saturn is seen to have moved in those four days is approximately 0.32° of arc, slightly more than for Jupiter over 2 days, again hopefully measurable.

Using these values, rₛ ~  9.4 AUs, while the accepted figure is around 9.6 AUs, which perhaps we could have achieved by averaging the results from many opposition events.

Again, checking Kepler's third law:

• the cube of its orbital radius is 9.4³ ~ 831
• the square of its orbital period is 29.45² ~ 867
• the former divided by the latter is ~ 0.96

Note: in no case has the estimate of the Solar Kepler constant ever exceeded 1!

Determining the orbital radius for Jupiter

Updated 27 April 26

Adapting the formula that we used for Mars, the radius of Jupiter's orbit, rⱼ, is given by:

rⱼ ~ (dₑ + sin θ)/ ((dₑ/pⱼ) + sin θ)

where pⱼ is its orbital period, for which we'll give the value of 11.85 that we found earlier.

Having said earlier that for Jupiter etc, it would be better to consider a larger Δt, for now, I'm going to use exactly the same value as for Mars, i.e. 2 days, and see how it goes, so as before, distance dₑ is 4π/365.25 AUs.

Considering the opposition of Jupiter on January 10, 2026 and using Stellarium to determine its position 24 hours before and 24 hours after, and this tool, θ, the angular distance Jupiter is seen to have moved in those two days, is approximately 0.27° of arc. This is smaller than was the case with Mars, now about half the angular size of the full moon, but perhaps still measurable with reasonable accuracy using the naked eye, a good star chart and a quadrant, say?

Using these values, rⱼ ~  5.1 AUs, which is very close to the accepted figure of about 5.2 AUs, which we would likely have arrived at by averaging the results from many opposition events.

Again, checking Kepler's third law:

• the cube of its orbital radius is 5.1³ ~ 132.7
• the square of its orbital period is 11.85² ~ 140.4
• the former divided by the latter is ~ 0.94

If we instead use the accepted value for the orbital radius, then this ratio becomes ~1, as we should expect.

For Saturn, the angle θ over just two days will be smaller again, so in that case it really will make sense to increase the time interval Δt and hence the value of dₑ.