The Sun's gravitational pull at the Earth's surface

We know from an earlier post that the acceleration due to a massive object is:

i. 4π²k/d²

where k is the Kepler constant for the massive object (yes, I'm avoiding G and gravitational mass) and d is the distance from that object.

Hence at the Earth's surface the ratio of the gravitational acceleration due to the Sun to that due to the Earth is:

ii. (kₛ.dₑ²)/(kₑ.dₛ²).

Where kₛ and kₑ are the Kepler constants for the Sun and Earth respectively and dₑ is the distance from centre of the Earth, i.e. one Earth radius, and dₛ is the distance from the Sun, i.e. 1 AU.

We know also that:

iii. kₛ/kₑ ~ 10⁶/3 &

iv. dₑ/dₛ ~ 1/24000

Hence, at the surface of the Earth, the ratio of the gravitational acceleration due to the Sun to that due to the Earth is about

10⁶/(3 x 24000²) ~ 0.0006.

This tiny, tiny acceleration, less than one-thousandth that we experience due to the Earth, is enough to keep the Earth in its orbit about the Sun, year in, year out!

At Saturn, ~ 9.6 AU from the Sun, the acceleration due to the Sun is about 100 times smaller than at the Earth and yet, it's sufficient to keep Saturn orbiting too!

Gravity is such a puny force and yet it permeates the cosmos!

The Astronomical Unit - improving on 8 light minutes

For almost as long as I can remember, I've known that:

• it takes light from the Sun about 8 minutes to reach us
• the Sun is about 93 million miles away
• light travels at about 186 thousand miles per second.

However, being on the trailing edge of things, it seems to have passed me by until just now that from the last two statements, we can see that in one half second, light travels one thousandth the distance to the Sun and so it takes 500 seconds for light to travel the full distance.

That's a nice easy round number to remember and is far more precise than the rather vague 8 minutes, which is out by twenty seconds.

Finding longitude before the invention of the marine chronometer

In recent posts about estimating the size of Astronomical Unit, I've stated that this couldn't have been done accurately before the advent of the marine chronometer to establish longitude.

Not so - it was the problem of the accurate measurement of longitude at sea, to determine the location of a vessel, that the marine chronometer solved.

Long before that, the use of astronomical tables, specifically of Jupiter's moons, allowed longitude to be measured with sufficient accuracy on land.

The relevant posts have been corrected.

Estimating the radius of the Moon (and the Earth-Moon distance)

Graphic to follow!

Many years ago I timed a lunar eclipse and attempted to work out the diameter of the Moon compared with the Earth's from that information. Unfortunately, I didn't keep a record, but was pleased to note that the result was very roughly in the right general area.

With an estimate of the size and knowing that the Moon is about half a degree of arc in diameter, one can then estimate the distance from the Earth to the Moon. This is what the Ancient Greeks would have done.

One of my assumptions was that diameter of the umbra was equal to that of the Earth. Now, I knew this to be incorrect, but I didn't bother to find out by how much.

(I also assumed that the centre of the Moon went through the centre of the umbra, also wishful thinking, but I'll leave that for now).

Here is a very belated attempt to calculate the size of the Earth's umbra at the Moon:

• let's think of a solar eclipse that is momentarily total
• let the Moon be at distance d from the Sun
• let the radii of the Sun and Moon be rₛ and rₘ respectively
• let the length of the Moon's umbra be lₘ
• we can say that lₘ = d.rₘ/(rₛ - rₘ)
• now think of a lunar eclipse.
• this time the Earth, radius rₑ, is at distance d from the Sun
• let the length of the Earth's umbra be lₑ
• we can say that lₑ = d.rₑ/(rₛ - rₑ)
• hence the ratio of the length of the Earth's umbra to that of the Moon is 

     lₑ/lₘ = (rₑ/rₘ).(rₛ - rₘ)/(rₛ - rₑ) ~ rₑ/rₘ ~ 3.7                    (since rₛ >> rₑ, rₛ >> rₘ)

Now, at distance lₑ, the size of the Earth's umbra is 0 and at distance lₘ (the Earth-Moon distance) the diameter of the Earth's umbra is therefore about (3.7 - 1)/3.7 or 0.7 times the diameter of the Earth. The difference is not insignificant and if we ignore it, we will over-estimate the size of the Moon and also the Earth-Moon distance. While I didn't take this into account, I'm sure that the Ancient Greeks would have done so in order to estimate the Earth-Moon distance, as they did so quite accurately, at around 60 Earth radii IIRC.

Earth 'weighed' against the Sun

Having discovered that the size of the AU can be estimated in Earth radii using the telescope, there remains one task: to convert Earth's Kepler's constant to AU³/year², so that we can it with those of the Sun, Jupiter and Saturn:

• 1 AU ~ 24000 Earth radii, i.e. the radius of the Earth is ~ 4.2 x 10⁻⁵ AUs
• we cube this: ~ 7.4 x 10⁻¹⁴ and multiply our Kepler constant (297) by it
• 1 day is ~ 1/365 years
• we square this: ~ 7.5 x 10⁻⁶ and divide the above result by it 

In Solar System units, the Earth's Kepler constant is therefore ~ 3 x 10⁻⁶, i.e. about one 300,000th the value of that for the Sun. Hence, we suspect (pre-Cavendish) that the Earth's mass is the same proportion of the Sun's, one 300th the mass of Jupiter and one 100th the mass of Saturn.

Further, given the Sun's angular size is about one half a degree of arc, we can deduce that its radius is about one hundred Earth radii and its volume about that of one million Earths.

Hence the density of the Earth is about 3 times that of the Sun (and of Jupiter).

Measuring the Astronomical Unit - part ii

Updated 29 May 26

Introduction

Here's a summary of two (identical!) estimates of the AU made in 1672, based on telescopic observations of Mars at or near opposition. They are remarkably accurate for the time, being well within 10% of the accepted figure.

With these estimates, for the first time the terrestrial scale could be connected to the astronomical, via the radius of the Earth, the size of which had been known since the time of the Ancient Greeks. 

(True, the Earth-Moon distance had also been fairly accurately known, but that was a dead-end as a unit of measurement).

Summary

i. Flamsteed (Britain)

Method: diurnal parallax, i.e. non-simultaneous observations made at the same location

Location: Greenwich (51.5 N)

Baseline not known by me, but:

• if observations were made at 9 pm then 3 am (90° Earth rotation later), then 

        baseline = radius of circle of latitude x √ 2 = Rₑ x cos 51.5° x √ root 2: ~ 0.9 Rₑ

• if observations were made at 8 pm then 4 am (120° Earth rotation later), then

        baseline = radius of circle of latitude' x 2 x sin 60°  =  cos 51.5° x 2 x sin 60 ~ 1.1 Rₑ

(where Rₑ is the mean radius of the Earth)

Pros:

• no travel involved
• no need to determine longitude
• one observer - consistency of observation

Cons: 

• need to allow for relative movement of Mars between observations

Result: 92.5 % actual

ii. Cassini, Richer (France)

Method: 'standard' parallax, i.e. simultaneous observations at distant locations

Locations: Paris (49N,2E), Cayenne (5N, 52W), separation 63° of arc (from here).

Baseline: ~ 1 Rₑ 

Cons:

• (very) long-distance travel involved
• need to determine longitude accurately*
• two observers needed - inconsistency of observation?

Result: 92.5 % actual

* In 1668 Cassini himself had produced astronomical tables in order to do just that.

Sources: wikipedia plus links above.

Comments

These estimates of the AU are remarkable achievements for the time - a mere 60 or so years after the invention of the telescope and a similar number of years since Kepler's theories.

However, it is inconceivable that two independent estimates using different methods should be in exact agreement! 

Flamsteed's method is ingenious, but I wonder how easy it would have been to achieve in practice. It surely must have occurred to Cassini also.

Further investigation is needed! 

Further reading

Six stages in the history of the astronomical unit, David. W. Hughes

The Mars Parallax Project, John J. McCarthy Observatory

Measuring the Astronomical Unit - part i

Updated 29 May 26

2004 transit of the Sun by Venus, projected on a
wall through one side of 10x50 handeld binoculars.

Having seen that trying to estimate the size of the AU in terrestrial units by measuring the Moon's penumbra during a solar eclipse is a non-starter, let's look instead at the tried-and-tested method of parallax.

Consider Mars, for example, when it is at opposition and about half an AU from the Earth. Let's create a baseline of 5 thousand miles, say, measured through the Earth, with at each end, an experienced observer and telescope pointing at the planet. Now arrange for readings to be taken at an agreed instant when the baseline is perpendicular to a line between Earth and Mars and thereby maximizing parallax.

The method needs:

• accurate and precise local time-keeping - only possible with the invention of the long pendulum clock in the late 1600s
• an accurate and precise measurement of the difference in longitude to determine the baseline and time-difference - actually possible on land with the use of astronomical tables before the invention of the marine chronometer.

If Mars were say 5 million miles away, then the angle of parallax would be one thousandth of a radian. One radian is roughly 60 degrees or 60 x 60 minutes of arc. One thousandth of that is roughly three and one half minutes of arc. That's about ten times the angular size of Mars itself. Now if the observers can draw Mars against the background of stars with reasonable accuracy, the difference in position should be measurably accurate.

If Mars were instead say 50 million miles away, then the angle of parallax would now be one tenth of that and about the same as the angular size of Mars itself. Perhaps this would be too small to discern. However, if the planet were seen to obscure a star (an example of an occultation), from one location and to touch it at the other, then that would provide a measurement of parallax.

Even if the angle were too small to measure, it should at least be possible to establish a minimum size for the AU.

Venus is closer to us at conjunction, just 0.28 AU away, a little over half the distance to Mars at its closest. Its maximum angle of parallax is therefore about twice that of Mars and hence easier to measure. How to observe it in daylight? Just wait for when it transits the Sun! Then during the transit, our observers can record the progress of Venus across the disk of the Sun, noting the times as they do so. Afterwards, they can compare their findings, noting the different positions of Venus with the same 'timestamps', allowing for difference in longitude.  Armed with this information and the length of their baseline (and its orientation at each instant to the line between the Earth and Venus), they will now have an angle of parallax for Venus measured against the Sun's disk. However, this will be smaller than the true figure, had they been able to measure it against the fixed stars. The reason being that there will also be some parallax for the Sun - 28% of that measured for Venus. So, the true figure for the parallax of Venus will be the measured figure times 100/72. While it sounds easy in principle, I won't underestimate the difficulty of doing this in practice, especially before the invention of photography.

The Geometry of a Solar Eclipse

Updated 23 May 26

Geometry of a Solar Eclipse

The diagram shows a simplified representation of a particular type of solar eclipse in which the Sun is obscured for an instant only. If the Moon were closer, the eclipse would be total for a period of time and if it were further away, the eclipse would be annular.

At right is a red vertical bar representing the Sun, radius r₃; in the middle is a green vertical bar representing the Moon, radius r₂; at left is a black vertical bar representing the extent of the observed  partial eclipse (penumbra), radius r₁. 

Point 1 represents the location of an observer (on the Earth) of this momentary total eclipse. 

The distance between point 1 and (the centre of) the Moon is d₁ (known); the distance between (the centre of) the Moon and point 2 is d₂ (unknown) and the distance between point 2 and (the centre of) the Sun is d₃ (also unknown). The total distance between the point 1 and (the centre of) the Sun is therefore d₁ + d₂ + d₃.

Marked, but in the end not used, are angles θ (half the angle subtended by the Moon at point 1), and φ (half the angle subtended by the Sun at point 2).

From similar triangles:    

1.    (d₁ + d₂)/r₁ = d₂/r₂ = d₃/r₃

and:    

2.    d₁/r₂ = (d₁ + d₂ + d₃)/r₃.

From 1:

3.    d₂ = r₂.d₁/(r₁ - r₂).

From 2 & 3:

4.    d₃/(d₁ + d₂ + d₃) = d₂/d₁ =  r₂/(r₁ - r₂).

So: 

5.    (d₁ + d₂)/ (d₁ + d₂ + d₃) = 1 - d₃/(d₁ + d₂ + d₃) 

=  (r₁ - 2r₂)/(r₁ - r₂). 

(Note:  r₁ > 2r₂)

Hence, the ratio of the Earth-Moon distance to the Astronomical Unit is (minus the radius of the Earth):

6.    d₁/ (d₁ + d₂ + d₃) = ((r₁ - r₂)/r₁).(d₁ + d₂)/ (d₁ + d₂ + d₃)

= (r₁ - 2r₂)/r₁.

So, all we need to do is measure r₁, plug it into the above equation along with the known value for r₂ and we are done! However, there is a problem. Let's say the ratio in the above expression is 1/n, then:

7.    r₁ = 2r₂.(n - 1)/n. 

This means we must measure r₁ with an accuracy (much) greater than 1/n - an estimate will not do.

I had hoped that this could form the basis of a simple method to estimate the AU, but that does not seem to be the case. And I've not even considered yet just how to estimate r₁ from the timings of a solar eclipse on the spinning Earth. Not simple then and I can only think that this was never a practical method either for finding the size of the AU. However, for high n, r₁ to all intents and purpose equals 2r₂, so this is instead one way that the diameter of the Moon could be measured.

Radio 4 on Long Wave

Updated 15 May 26

The BBC has recently announced that transmissions of Radio 4 on Long Wave will end soon, after 100 years of the BBC broadcasting on this wave-band. The decision has been expected for a long time and other than an exact date is therefore no surprise.

I recall listening to the news on Radio 4 in France with my parents in the 70s and many residents in Europe must have relied upon this service in times past for impartial news from a much-trusted source (those were the days!).

As recently as around the turn of the century, I was also able to pick up several overseas stations on this wave-band. Now it's just Radio 4 and soon, sadly, all will be silent. Hopefully, Medium Wave will survive a bit longer, even with fewer stations being on-air each year.

A 1970s School 'Computer'

Originally posted on the RetroMat website, but fits better here.

Programming at school in the decade before the microcomputer revolution.

Diehl Combitron S with paper tape peripherals - a machine of similar type to the one we used.
Picture courtesy of technikum29.

In around 1973 or '74, at Wellsway School in Keynsham, our Maths class was introduced by our wonderful teacher, Mr. Grace, to a machine like the one above. It was shared between three schools in the area, one term at a time. While it was referred to as a 'computer', it was really a desktop calculator with some programming capability. We had taken Maths O-level a year early (a reflection of the SMP syllabus) and in the fifth form we had a little more time on our hands for such things. Programming was a new experience for us. Indeed, when we were first introduced to the machine, simply using a calculator would have been novel, with pocket calculators being a recent development, hence expensive and comparatively rare.

Programs were worked out in advance and written down by hand on paper. Then when we had our allotted sessions on the machine, we'd key them in and try to get them to run. If we were lucky, we'd correct them there and then. Either way, they would then be saved onto punched paper tape (possibly manually). As this was a shared machine, getting a program to run successfully was time-consuming and I remember getting up early on more than one occasion to get some time on the machine.

I recall that programs comprised up to 100 programming steps*, contained in 10 program registers of 10 steps each. If you needed to continue after 10 steps, you would go to the next program register. Perhaps we could implement subroutines**, maybe without using that term, using these program registers? In addition, there were 10 registers available for storing numbers. It was possible to repeat steps by jumping back to the beginning of a section of code and - crucially - to execute code depending on some condition being met. This last features provided the ability to extend programming beyond being simply a series of automated steps on the calculator (keystroke programming).

As to the device itself, I had long forgotten the name. However,  I did recall the distinctive symbols used for sending and retrieving data to and from registers, like this  -  V  - for one such operation and the same symbol but upside down for the other operation.  After a lot of searching, I found that these symbols seem to be unique to German manufacturer Diehl, and thanks to information about the range of their calculators on the very extensive CALCUSEUM site, I think our machine was either the Combitron-S, shown in  the centre of the picture above, or a similar machine, the Combitronic, these devices being limited to 10 program registers of 10 steps each. However, neither the manufacturer's name nor the names of the machines sound at all familiar, so I wonder if it could have been re-badged with a different company's name?

Assuming I'm right, then our machine dated from the late 60s. As I wasn't aware of its existence before 1973, I wonder if it might have been donated by an organization replacing it with a more recent machine?

(BTW, the tape puncher at the left in the above picture seems to have been an automatic one that punched as you typed on the main machine. I don't recall using one like that - it was likely this manual tape puncher here that we had).

Assuming it was this machine, it was pretty basic - no stored values for π or e and no trig. functions. (Even my first pocket calculator, bought a year or so later, had functions that weren't available on it!)  However, there's a sample program (via one of the links below) that calculates e(x) using a Taylor series in just 4 program registers, using 2 memory registers. No doubt sin (x) or cos (x) could be done similarly. 

It was also quite clunky to use - requiring numbers to be moved in and out of the central unit, to and from registers. While cumbersome, it was however a great introduction to programming in machine language for me, specifically in z80 Assembly Language for the Sinclair Spectrum some ten+ years later!

Set exercises included e.g. calculating the volume of a sphere based on a radius input by the user (π would have been assigned to a store by the programmer), playing the game of NIM against the machine and simulating a solution to the Tower of Hanoi puzzle. That first example would have used simple keystroke programming, the last two would have used logic. My code is long gone, but I was inspired to write NIM for the Sinclair Spectrum in recent years!

After completing these challenges, I decided to set my own by using an approximation method I had recently learnt about to determine the square root of a number. Never mind that the 'computer' came with its own, much faster, square root function, nor that there were likely many more suitable candidates for calculating a function! It was great fun and quite satisfying to do. The program looped to determine successive approximations of the root and was supposed to stop when they were the same. However, sometimes it continued to run with the printed approximations oscillating between two values and the program had to be stopped manually. Again, the code is long gone, but here is my Sinclair Spectrum version.

An enterprising fellow pupil used the machine to generate artillery tables and these were displayed in a chart along with the relevant equation at a stand at an evening event for parents of prospective pupils. Trigonometric functions would have been needed for this, so he must have coded his own!

By the time I left the school in 1976, the machine seemed very dated. In fact, I remember the Head of Maths, Mr. Batey, getting a programmable pocket calculator not long before I left and our machine must surely have been ditched soon after!

(Not that I was aware at the time, but Diehl had been active in updating their range - the Alphatronic from 1973 was much more powerful, could be connected to peripherals with magnetic media, had alphabetic keys (presumably so that text messages could now be displayed in programs), and 'proper' programming, including keys for GOTO, GOSUB and LABEL. Later models were even referred to as 'computers' by Diehl. In the late 1970s, the company released a mini-computer range, the DS2000 and DS3000, with integrated CRT display, aimed at the technical and scientific market, but it seems they left the business soon afterwards).

It would be great to see one of these machines working again. And I wonder - how difficult would it be to create an emulator?

(Almost) finally - if any fellow ex-pupils or ex-staff (!) at Wellsway, or one of the other schools in the area, remember this machine, I'd be delighted to hear from you!

Foot notes

* actually it's just 10 characters per program register, so even more limited than I had remembered!

** I was wrong about subroutines, if our machine was indeed a Combitron-S or Combitronic. Having reviewed the sample programs (see below), it seems that while you can jump between program registers, there is no hierarchy and you cannot return control to a calling program register.

More information

• For sample programs for a similar model, albeit with with less powerful programming capability, the Diehl Deltronic P, see the links to the brochure pages (in German) here
• To make sense of the syntax used to perform calculations, it may help to view this schematic (in French) for the Diehl Sigmatronic calculator (not programmable as far as I can tell, but the calculator part seems to be the same). In particular, note that there are separate registers for addition/subtraction, multiplication/division (as well as for accumulation). 
• For more information on the range of Diehl electronic calculators, see here and this document (in German) here
• For a closer look at one of the sample programs above, and the calculator syntax, see this document for some comments I have added
• Summary of symbols used by Diehl electronic calculators (focus on Combitron-S, Combitronic)
• An attempt to understand the format of the punched paper tape. (If only I had kept my programs then it should have been possible to decipher them simply by looking at the locations of the punched holes!)

Ferguson Tidal Clock

Originally posted on the RetroMat website, but fits better here.

Source: The Antique Clock Company

In the August 2022 edition of Clocks Magazine, there is an article by Brian Loomes about the astronomical clocks with tidal indications of Phillip Lloyd based on a mechanism devised by astronomer James Ferguson.

One of the clocks shown in the article - the Hotwells one - was for sale as at August 2022 (and still seems to be so as at May 2026). The dial is a thing of great beauty, although perhaps not intuitive to understand.

There is a very good explanation of the dial and mechanism used by Ferguson, albeit in dated language, in the Encyclopaedia Londinensis published in 1811, which I've copied into this document for ease of reading, with some alterations:

• the low-resolution black and white diagrams accompanying the text have been replaced with high resolution images of the original colour plates by J. Pass, 1809, from the Wellcome Collection
• annotations have been made to one of the figures and some footnotes added
• updated spelling of words where the archaic long 's' is used.

With regard to the Hotwells clock by Lloyd: the indicator used by Ferguson to show high water is not present, perhaps having been replaced by the pair of 'spikes' on opposite sides of the central disk representing the Earth. If this is so, then because these are offset to the 'tidal bulge' around the Earth, this depiction corrects the misrepresentation of high water in the Ferguson clock.

However, such a device would seem to be less accurate in use, being some distance from the dial on the sun plate.

Also, the Hotwells clock has two additional, centrally-mounted hands. Presumably the straight hand is for showing seconds, with the other for minutes, the hours being indicated by the short hand with the sun symbol mounted on the sun plate.

Plimpton 322 clay tablet

Originally posted on the RetroMat website as part of a page on programming the Sinclair Spectrum to generate Pythagorean Triples.

Source: wikipedia

Plimpton 322 is a Babylonian clay tablet containing a list of triangles composed of Pythagorean Triples, and their associated properties, in the form of a table with 4 columns and 15 rows ordered in descending sequence of (short side/long side) squared. 

These triples can be generated by using a method, such as the one we know as Euclid's Formula, and the following analysis refers to using that formula. Presumably the Babylonians used a similar method.

There are 15 triples implied on the tablet (it lists the short sides and hypotenuses, but not the longer sides). Of these, 13 are primitive. Eight have c > 1000 and the largest has a = 12709 and c = 18541. These are large triangles! (To generate the largest, m=125 and n=54). 

(I wonder what method they used to determine that these are co-prime and if they knew about what we now call Euclid's Algorithm? (Not to be confused with the above similarly-named formula!))

The tablet is broken and it is possible that there are missing columns at the left. Angles aren't used, but can be determined. If we define the angle as being next to the long side and hypotenuse, then angles* range from ~45 degrees at the top of the table to ~32 degrees at the bottom. The difference between the angles associated with successive rows is just under one degree on average, but actually varies from less than half a degree to nearly two and there seems to be no pattern to this.

Why were those particular triples chosen? I found that by increasing the maximum value for m to the 125 needed for the largest triangle in the tablet, the formula generates 441 triples in this range, providing many more suitable candidates for a more evenly-spaced table.

Why this range of angles? Perhaps it is one of a set - four such tablets could cover angles from 0 to 45 degrees (albeit with 60 rather than 45 entries!). For 45 to 90 degrees, the triples could be reused, switching the short and longer sides.

Was the tablet intended to be used as a trigonometric table, as we understand that term, with angles being present originally in a column now missing? (Presumably, angles would have had to be obtained by direct measurement of suitably scaled-down triangles). If so, it predates those of the Ancient Greeks, who are usually regarded as having developed the first ones, by 1500+ years! However, such a table would have been of limited practical use:

• the angles, if present, would not have been 'tidy' integers or rational numbers 
• there is an uneven difference in angle between successive entries
• the number of entries (15) also counts against (would have expected 14 for the range given).

Perhaps instead they were searching for triples that could produce (near) whole integer angles and this was a work in progress, a prototype of trig. tables to come?

Regardless, what I find truly incredible, perhaps even more than the information it contains, is that in spite of its great antiquity, it is in a form that we recognise today - an ordered numbered table consisting of rows and columns!

* angles of the triangles associated with the triples: 44.76, 44.25, 43.79, 43.27, 42.07, 41.54, 40.32, 39.77, 38.72, 37.44, 36.87, 34.98, 33.86, 33.26, 31.89.

Earth's Keplerian system & towards universal gravitation

Updated 31 May 2026

Having investigated the Solar, Jovian and Saturnian Keplerian Systems, let's take a closer look at home and start with trying to determine the value of its Kepler 'constant' (should it exist) by looking at the Moon's orbit about the Earth.

It's at a distance (measured with reasonable accuracy by Ptolemy nearly 2000 years ago) of about 60 Earth radii and takes about 27 days to complete one orbit.

From this we can say the value of the Kepler 'constant' for the Earth is ~ 60³/27² ~ 296.

In the case of the other systems, there are many orbiting bodies and we can calculate an average, but in this case, if we go back many decades at least, there is only one such body orbiting the Earth.

However, let's take inspiration from Newton, who realized that an apple falling to the ground also experiences an acceleration towards the centre of the Earth. This acceleration is commonly referred to as g and can be measured accurately by means of a pendulum, length l, the period of which is 2π√(l/g).

We can generalize the formula from the previous post and say that the acceleration at distance d, regardless of whether the object is orbiting, is:

4π²k/d².

At the Earth's surface, we state in this case that d = 1 (the same as if all the Earth's mass were concentrated at its centre) and g = 4π²k, where k is the Kepler 'constant' for the Earth. If we know g, we can therefore determine that:

k = g/(4π²).

In metric units g ~ 9.8 m/s² and we need to convert this to the units used when calculating the value from observation of the Moon, i.e. Earth radii/day²:

• the radius of the Earth (also measured, albeit in different units, with reasonable accuracy by the Ancient Greeks) is ~ 6.4 x 10⁶ m and we divide our value of g by this number
• there are 86400 seconds in a day, squared is ~ 7.5 x 10⁹ and we multiply the above result by this number.

In these new units g ~ 11719 and dividing by 4π² gives ~ 297, which is very close indeed to the value we derived earlier.

We can conclude therefore that the Earth, as with the Sun, Jupiter and Saturn also has its own Keplerian system, where the central body exerts a gravitational force that follows an inverse-square law.

From earlier considerations, that force appears to be related to the mass of the central object (Sun, Jupiter etc), possibly simply proportional to it, in which case the Kepler 'constant' for the system is also proportional to the mass of the central object.

We can surmise therefore that all massive bodies, including those bodies without satellites of their own, exert such a gravitational force on surrounding matter. (Surface features on the Moon, for example, and the near spherical shape of all planets, including those without moons, provide strong evidence for this).

This was to be confirmed experimentally by Henry Cavendish in the laboratory at the end of the 18th century - an incredible achievement for the time given the weakness of the gravitational force. 

From Kepler's third law to the inverse-square law

When a small object orbits a massive one, such as a moon going about its planet, say, it experiences an acceleration towards it. In perfect circular motion (an approximation), that acceleration is: r⍵²,where r is the radius of the orbit and ⍵ is its angular speed, which is 2π/t, where t is its orbital period. We know this from Newton.

We know also from Kepler's third law that

r³/t² = k,

where k is the Kepler 'constant' for that system. 

Hence the acceleration 

r⍵² = r4π²k/r³ = 4π²k/r²

This is the famous inverse-square law for the gravitational force (also from Newton of course).

The Saturnian Keplerian system

Decades after the discovery of Jupiter's moons, Saturn was observed to have its own satellites: the first being Titan, the largest, in 1655.

Let's try to compute the Kepler value for the three largest, using the same method as for Jupiter, noting however, that accuracy and precision would likely have been low in the early days, so the figures below (especially its density) may not have been achievable until much later.

There is no orbital resonance with these, so the table shows the period in days.

(Rhea's orbit is actually about 4.5 Saturnian diameters. Were we able to estimate to this precision, then we should expect the Kepler value to be ~4.5).

We see that the values in the last column suggest a Kepler 'constant' for Saturn of ~ 4 and hence that Saturn, as well as the Sun and Jupiter, also has its own gravitational system obeying the same laws.

Converting this to Solar system units:

• as Saturn's angular diameter at opposition is ~ 1/3 minute of arc and it's 8.4 AUs away, it's diameter is ~ 0.0008 AUs
• we cube this:  ~ 5 x 10^(-10) and multiply our Kepler constant by it
• 1 day is 1/365 years, so we square this ~  7.5 x 10^(-6) and divide the above result by it

Hence, the average Kepler 'constant' for Saturn is ~ 0.0003.

We deduce that its gravitational pull is ~ 0.0003 times that of the Sun and ~ 0.3 times that of Jupiter. 

We suspect its mass is also in the same proportions. That being the case, as we know its diameter is ~ 0.8 of Jupiter's, its volume is therefore around half of Jupiter's and its density ~ 0.6 times Jupiter's, indicating a different composition.

Moon	Orbital radius (r) in Saturnian diameters	r^3	Period (t) days	t^2	r^3/t^2
I. Rhea	4	64	4.5	20	3.2
II. Titan	10	1000	16	256	4
III. Iapetus	30	27000	79	6241	4.3

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