Sunday, November 1, 2026

A formula (Madhava/Gregory/Leibnitz) for Pi

The Taylor series for arctan (1) is:

1 - 1/3 + 1/5 - 1/7 ....

Since tan 45° is 1 and 45° is equivalent to π/4 radians, multiplying the above expression by 4 will yield π itself.

(I suppose we could also look at 6 arccos (0.5), but the first is better, because the argument is simply 1).

The trouble is, just look at the denominators: 2n-1 for the nth term. That means it's an incredibly slow convergence as shown on these screenshots from a program running on an emulated Sinclair ZX Spectrum. (The terms are numbered from 0 in the program). In the first three, we see the first, second and third decimal places having stabilized at their true values after 24, 626 and 2456 steps respectively. Imagine trying to do this by hand! 

The fourth screenshot shows the output from a modified form of the program which prints every thousandth result. As these are extracted results, we cannot say that the first appearance of a correct decimal place value is a stable one. For example after 2000 iterations, we have 3 correct decimal places, but we know the third is not yet stable. At ~11000 iterations, we seem to see stable results for the fourth decimal place, but even after 22000 iterations, the fifth is still way off (4, not 9), as is indeed the case (not shown) after another 22000 (6)!

I love the way the most fundamental of irrational numbers can be arrived at by such a simple series of rational numbers, even if the method is glacially slow!





Saturday, October 31, 2026

Across the Universe

Updated 1 Nov 26

I have recently come across a couple of items online based on the elapsed time for occupants of a spaceship accelerating at 1g to traverse cosmic distances.

First a video, Something Terrifying Would Happen If a Spaceship Constantly Accelerated at 1G Through Space..., on YouTube channel ExoSci, posted 27 Sept. The audio refers to an article, but which? Could it be this one, How long does it take to travel through the Universe at 1g?, posted by Ethan Siegel on Big Think just the day before? Too soon surely, but perhaps Siegel had posted something similar earlier. Astonishingly, he concludes: All destinations within the reachable Universe are achievable in 25 years, maximum. Of course, that is for the occupants of the spaceship and is a consequence of relativistic length-contraction over the cosmic distance travelled.

For some reason, this made me remember an amazing fact I first came across many years' ago - if the Earth were to be squeezed as a cylinder of 1 square metre cross-section (maintaining its density), its length would equal the diameter of our galaxy! 

Let's check:

  • the radius of the Earth is 6.4 million metres
  • therefore its volume is ~ 10^21 cubic metres
  • with a cross-section of 1 square metre, the cylinder would have a length of 10^21 metres
  • one light year is about 10^16 metres, so that makes the squeezed Earth 100 thousand light years in length
  • our galaxy is believed to be 100 to 150 thousand light years across, so the assertion is roughly correct!

Why stop there? What if we were to reduce the cross-section of the cylinder to 1 square centimetre? There are 10,000 (100 squared) of those in a square metre, so the squeezed Earth would now be 10,000 times longer than before, so 1000 million, or 1 billion light years in length!

What if we were to reduce the cross-section of the cylinder further to just 1 square millimetre? There are 100 (10 squared) of those in a square centimetre, so the squeezed Earth would now be 100 times longer than before, so 100 billion light years in length! That's about the diameter of the observable universe!

(We could also dial back and increase the cross-section to 1 square kilometre, in which case, the cylinder would stretch to about one millionth the diameter of the Milky Way, i.e. to one tenth of a light year, about the distance from the Sun to the inner edge of the Oort Cloud).

Conclusion: the (observable) universe is vast, but not unimaginably vast. (And 'just' a million Milky Way galaxies placed side-by-side will span it).

Thursday, October 15, 2026

A 100-year-old crystal radio

Updated 27 Oct 26





Crystal radios are wonderful, able to pick up broadcasts without consuming energy. Old technology is cool, so that makes this antique crystal radio doubly so.

I was very kindly given it by a friend of my mother's. It had belonged to her grandparents and the (near pristine) licences that came with it are for 1923-24 (an interim licence costing 15 shillings) and 1924-25 (a receiving licence costing 10 shillings), both for an address in Balham and for 1929-30 (another receiving licence still costing 10 shillings) for an address in Merton Park, both in South London. 

That 15 shillings in 1923 is worth nearly £60 today. According to the weekly Popular Wireless magazine of 13th Oct. 1923, there was a deadline to register previously unlicenced radio sets (as they were, and sometimes are still, called), including home-made ones, by 15th Oct. - see here, page 10 - by means of the newly-introduced interim licence. On the document, it is described as a Licence to establish a receiving station and was presumably a one-off, standard receiving licences (also known as B.B.C. licences) being payable in subsequent years. (It seems that new home-made radio sets were to be covered by a newly-introduced constructor's licence, also costing 15 shillings).  I think we can deduce that the radio, whether manufactured or home-made, had been in use for a while and the licence, dated 9th Oct., was purchased with the deadline looming. The local radio station at the time was 2LO, set up by Marconi in 1922, but taken over by the newly-formed B.B.C. later that year. It broadcast from the Strand in London, so presumably reception was good just a few miles away in Balham. I suspect that the radio was bought in 1922 or '23 for the purpose of picking up that station for entertainment etc. and that the owners were early adopters of the new medium, albeit as consumers rather than hobbyists. 

The wooden chest looks to have been made by a competent wood-worker. A commercially-made case would likely have a manufacturer's mark and instructions and possibly licencing information as well and be varnished, while this one has neither mark nor text and is stained. In order to accommodate the controls, a deep flat-topped lid would have sufficed, but instead, the builder has opted for a pitched roof type of construction with semi-circles of wood over the 'gable ends', perhaps to give a first impression that the whole top is curved. The lid won't quite close now, but perhaps some adjustments to the hinges will fix that.

The panel is of a black plastic material (ebonite?) and has a typical layout for such radios of this period. It sits on a ledge in the cabinet and can be lifted out. There is no support in the middle and it has bowed downwards there a little over the years. The space below holds the coils and wiring attached to the underside of the panel and also serves as storage, in this case, for the licences already mentioned, and for the detecting crystals. There is no maker's mark on the panel either, but that seems to be typical even for the manufactured ones. It could have been sold as a kit, or perhaps as a fully-assembled item ready to be put into a cabinet. 

Detection is by means of a semi-conducting point contact between a wire - the cat's whisker - and a crystal (galena?). There are several crystals: two loose larger ones were in the storage compartment, as well as two boxes: one small, marked Talite at the base, containing several small crystals, and one larger, empty, marked Eiffel Tower Brand. 

(It's funny to think that while the crystal detector soon gave way to the diode valve detector, a few decades later this in turn gave way to the crystal detector once more, albeit in a smaller, packaged format, as can be confirmed by looking at a later germanium diode with a magnifying glass. The big advantage with the new types of detecting diode was that it was no longer necessary to adjust the cat's whisker wire, or to try different crystals, in order to get a radio to detect optimally. Indeed, when I made a crystal radio in the 1970s, once the diode/detector was installed, it wasn't given a second's thought).

Tuning is via two controls: one at right  (coarse tuning), which selects various tapping points on the main coil and one at left (fine tuning) which turns a second coil placed at the end of and just within the main coil. I believe this is known as a variometer. Thus the radio is tuned by changing its inductance, unlike later types that were tuned by varying the capacitance.

The tuning knob at left is missing and was presumably the same as the one at right. The pillars on which these were mounted have screw-threads and are 3/16 inches in diameter. I've not been able to remove the knob at right. If it was screwed down onto the pillar, it would surely have had to have been secured by glue, but since thread-locking glue seems to be a more recent invention, I'm rather puzzled.

Interestingly, the output terminals are marked TELEPHONES rather than HEADPHONES. From the copy of Popular Wireless mentioned above, this seems to have been a common alternative term at the time. Indeed, in the first edition of the magazine in June 1922, it refers to the national wireless telephony broadcasting scheme about to begin operations! Presumably this description was an adaptation of that of so-called wireless telegraphy for Morse code over radio, but really it is better applied to today's mobile phone network.

Also present are two stranded cables, one coated in fabric and the other in a sort of rubber which is now crumbling a little. Both still conduct electricity. Neither is more than about one and a half yards in length and they were presumably for connecting the set to terminals a short distance away, possibly indoors, for aerial and ground.

I have tested it, although with a far shorter aerial than it deserves. For comparison, I used the same aerial and ground connections with a more recent educational crystal radio, by Maxitronix. There are no headphones and as I no longer have working crystal earpieces, output was to a high-gain LM 386-based amplifier and a speaker. With the Maxitronix, I was able to choose from three points to connect the aerial to optimize reception. During the afternoon, both radios were swamped by presumably Far Eastern stations, some/one of which were in English. I think this may be because the metal-framed window in the room pulls in these stations. IIRC, I've picked up these stations before - some may even have been short-wave ones. The antique was much the quieter of the two radios. On the Maxitronix, I was also able to pick up BBC Radio 5 Live at reasonable volume. At night, I was able to pick up this station on both radios, but the antique radio was again far quieter. On this radio, getting a good working contact between cat's whisker and crystal could be a little tricky. Also, sometimes the radio went silent and playing with the cat's whisker made no difference. I don't know if this was because of poor contacts elsewhere, or because of incompatibility with the amplifier, or a combination of both. I would have liked to have bypassed the detector with a germanium diode to compare performance, but there is no suitable mounting point and even if I had found one, no doubt getting a good connection would have been a problem.

It's impressive that such an old radio still works today, even if it is on the quiet side! No doubt, if I stripped it down, cleaned all the contacts, re-soldered the connections and attached a longer aerial, performance would be far improved. However, I think it's better to keep it in original condition as far as possible and just display it. And as AM radio stations are in terminal decline, in Europe at least, there also seems little point. Perhaps, I could clean the brass though...

According to an article How to Make a Loud Speaker in the third edition of Popular Wireless magazine, for shared listening, a second pair of headphones could be added (in series) to a radio. Alternatively, if close enough to the transmitting station, a telephone earpiece could be pushed against the soundbox of a gramophone player and the sound could then be heard from its horn, albeit quietly, all without amplification! This radio could therefore have seen many years of useful service, as is suggested by the presence of a licence for 1929-30. At some point though, an all-in-one valve set, being simpler to use and not needing a long* aerial, would have been increasingly difficult to resist and so the old radio would have been consigned to an attic or basement, perhaps.

*In the first edition of Popular Wireless magazine, there is an article on constructing an outdoor aerial, which is obviously aimed at users of crystal receivers in the main, but seems to suggest that owners of valve radios could also benefit. However, since valve radios could likely receive well enough for most listeners with just an indoor aerial, I suspect that in most cases, only if an outdoor aerial like this had already been constructed for use with a crystal receiver, would it have been connected to a valve receiver. The article says: For the best results the aerial should be a fair height from the ground - say, not less than 40 feet, whilst its overall length, in order to comply with the Post Office regulations, must not exceed 100 feet. It also says: for practical purposes 100 feet of wire will suit the average amateur quite well to begin with! (The exclamation mark is mine, btw). If there wasn't space for a full 100 foot length aerial, then the article says: a twin aerial is a good alternative... In this case 140 feet of wire may be used, although it's not clear to me where that number comes from. That 40 foot height wouldn't have been possible for a typical two-storey house, 25 feet maximum being more realistic, if the aerial was to be run from the top of the chimney, as shown in the article with photos of a determined-looking chap doing just that, with no hard hat and a pipe clenched between his teeth! Even then, it would have been easier and a little safer to run it lower still from the eaves, as shown in the 1970s Ladybird book making a Transistor Radio, which has a section on crystal radios. 

Monday, October 12, 2026

The LGP-30: a 70-year-old mini computer

Updated 13 Oct 26

This early small digital computer (introduced in 1956) and its manufacturer, Librascope, have somehow escaped my notice until just recently, following the widely-reported death of pioneering software engineer Margaret Hamilton, whose first work with computers was on one of these machines at MIT, as she recounts here. Nor did I know that such small-scale computers existed so early on!

Of particular note: it has drum (not ferrite core) memory comprising 4096 words of 31 bits each (so, 124k bits equivalent to 15.5k bytes in modern computing); uses solid-state diode logic, but also vacuum tubes; has just 15 flip-flops; has built-in multiplication and uses an oscilloscope to display information about its registers.

Also, of interest to me, is that its designer, Stan Frankel, whose work on the Manhattan Project sparked his interest in electronic computation, also designed the Combitron desktop programmable calculator by Diehl of Germany, (almost certainly) the first programmable machine that I ever worked on.

Source: wikipedia


Friday, October 9, 2026

SAP-1 Simulator

Updated 10 Oct 26

I wrote a while ago about a home-built 8-bit computer, the SAP-1, with an example of a simple program to add together the contents of two memory addresses.

I was thinking about this again earlier today and wondering if anyone had written a simulation of the machine. Well, I've found one of sorts here, where ones for the SAP-2 and for a Turing Machine can also be found.

Why 'of sorts'? Because it's high-level, so we don't get to see the microcode beneath the instructions. This also means you can't add to the instruction set (something that can be done on the original machine), so unfortunately there's no support for looping (jump) nor for logic (conditional jump), both of which are really essential requirements.

Here's a simulation of that program mentioned earlier, with the decimal values 28 in address 14 and 14 in address 15 and the output decimal 42 after running it. (I'm not sure about the decimal 14 in the accumulator though). See the first link above for a detailed breakdown of the microcode steps involved.

Thursday, October 8, 2026

Heavy breathing from the woods!

More strange noises to report! They are like this and come from nearby woods at night. They have been heard on many occasions - quite un-nerving the first time! Presumed to be made by stags as there are deer there. No specific time of year, as I have several recordings dated January to June.

(The recording may be on the quiet side if played online, in which case, download it and try again).

Nocturnal screeching!

Recording made around 11 pm towards the end of June this year, just one of several occasions when there have been these screeching sounds outside our house. I don't recall seeing anything, perhaps just catching the occasional fluttering as well. I'd heard such sounds in previous years and had always thought that they were made by bats, while also being fully aware that their echo-location uses ultrasound. Once disabused of the notion that bats also make audible sounds, I surmized that these must have been made by birds instead, but which ones? I was amazed that the BirdNET app identified them as from Tawny Owls! We often hear typical owl hoots, living close to several woods. Almost certainly though, these sounds are made by owlets calling out to be fed.

(The recording may be on the quiet side if played online, in which case, download it and try again).

Non-magnetic stainless steel!

I hate to throw out old motors etc. without first trying to retrieve parts like wire or magnets. The item I was attempting to dismantle was an old water pump for a garden water feature. It was completely sealed and looked to be difficult to break into, so I didn't try, but I did save a magnet which was outside the main body. It was dirty, so I washed it and then placed it onto the draining board section of a stainless steel sink. I expected it to snap into place, but nothing happened, which was a great surprise! It turns out that the crystalline structure of austenitic stainless steel renders it non-magnetic. Amazing to have discovered something so basic so late in life!

Monday, June 29, 2026

A Visual Model of Relativity

Updated 22 Sep 26

I've updated and re-instated this document on the RetroMat website. It goes from Special Relativity to a specific case from General Relativity using a simple visual model, some reasonable assumptions and relatively (!) simple maths, with not a tensor in sight!

Monday, June 15, 2026

'Upgrading' a Linux PC

A departure from my usual retro theme...

Until last year, I'd been happily using a laptop bought in 2008! It had been shelved for many years and when I retired was brought back into service, but running Lubuntu, a lightweight distribution of Ubuntu Linux for the desktop. Incredibly, it served me well for about four years.

Last year, my wife replaced her laptop, so that she could run Windows 11. It was a mere(!) 7 years old and had (and still has IMO) a decent spec., but Microsoft had decided they weren't going to support it, so I took it over. Lubuntu were abandoning the lightweight aspect, so I decided that I may as well use another distribution and settled on Linux Mint instead, another Ubuntu variant.

It was OK, but the mouse pointer was often replaced by a spinning icon, indicating the system was a bit too busy to devote itself to my needs. There was the odd glitch and Ubuntu itself is now becoming resource-hungry, with a (crazy, IMO) minimum memory requirement of 6 GB, so I decided to replace it and chose Fedora Workstation.

I started and seemingly made reasonable progress at first. True, it was a faff trying to boot from a USB key with the new OS, but that's BIOS (or UEFI) for you. (A further complication for me is that not all USB ports on my laptop can be used for booting).  The OS seemed to get installed, but I was left with a simple screen, no menu and just (so I thought) the ability to alter some settings by right-clicking the mouse. I rebooted the machine, thinking that would fix it, but was left with the same thing. In desperation I clicked on the top left of the screen which showed a sort of dash and a dot that I'd never seen before. Doing this then revealed the 'apps' with which to use the system - not a menu in sight! (Now, perhaps I should have read the documentation, or been a little more inquisitive, but I genuinely thought there was a problem). What a crazy decision to minimize everything on a new installation, where there is a high chance that the user, like I was, is new to the UI. It seems to be a triumph of aesthetics perhaps over common-sense and what a poor introduction to the UI (Gnome)! After several hours of playing around, I really hated it anyway and installed the KDE UI, which I had been using with Linux Mint. I did so, but couldn't fully get rid of Gnome, so decided to start again and to install Fedora KDE Plasma Desktop instead!

At this point, I discovered the fun was only just beginning! The main problems I encountered with this new OS:

  • Whilst I'd been able to write the Fedora Workstation iso file to the USB key OK from Linux Mint and boot from that without too much difficulty, trying to do so for the new OS on Fedora Media Writer was a problem -  I just couldn't boot from the USB key at all. It seemed that Legacy (BIOS) booting was no good and I had to use UEFI instead.
  • After that, it booted from UEFI, but then it couldn't see the internal disk on the laptop! So, after much digging around, I found out that I had to change SATA operations to AHCI from RAID in BIOS/UEFI.
  • I took the option to overwrite the existing Fedora installation, but that failed, so I chose to write to the entire disk.
  • Finally, having installed the OS, I then installed Chrome and encountered a weird error with it. The issue was that while I could log in to my Google account, I couldn't see my bookmarks. Searching the issue came up with lots of helpful advice about synching the bookmarks, but I just couldn't see an option to do that. After, much digging, I had to enter chrome://sync-internals into the browser, then enable something called Sync-The-Feature. I got some pop-up about security and after that, the synch options became available and I was able to finally see my bookmarks! A pure Chrome issue or a combination of Chrome and new OS, I wonder?

I had a career in IT and am a reasonably competent user. Even so, I estimate it took at least 8 hours for me to do the 'upgrade' to Fedora. The experience puts me in mind of the difficulties I used to encounter decades ago, when nothing was ever simple with PCs and you needed to be a real 'tecchy' to get things working. Since then a lot has changed and we should expect things to work with minimal fuss. Software - OSs, UIs and applications - should be lightweight, intuitive, robust and not require extensive problem-solving.

As to what happened, perhaps part of the problem is that there are too many flavours of Linux and UI out there and they have to work across a wide range of hardware. It's all very fragmented, as indeed was the Unix world before Linux. Maybe things would work better if there was less choice?

(Oh and by the way, simple copy-and-paste doesn't seem to work with the mouse on my new system!)

Wednesday, June 10, 2026

Scratch dials - a closer look

Updated 11 June 26

This replaces an earlier post.

As noted in the previous post, between the Spring and Autumn equinoxes, such dials, being South-facing, are in shadow nearer sunrise and sunset. Let's use Stellarium for a few examples at 51 degrees North:

- at the Summer solstice, the Sun enters the Southern half of the sky (i.e. lies due East) 3 hours and 40 minutes after sunrise, when its altitude is about 30 degrees and the dial therefore indicates the end of the second hour and the start of the third (no good if we need to know when the second hour starts and similarly for the final hour); the dial is illuminated by the Sun for just over half the daytime,

- about the start of Summer, May 4th say, the Sun enters the Southern half of the sky 2 hours and 20 minutes after sunrise, when its altitude is about 20 degrees and the dial therefore indicates part way through the second hour (again no good if we need to know when the second hour starts, but perhaps ok for the start of the final hour, which can be guessed at); the dial is illuminated by the Sun for just over two thirds of the daytime,

- even in mid-April, less than 4 weeks after the equinox, the Sun enters the Southern half of the sky 90 minutes after sunrise, when its altitude is about 13 degrees and the dial therefore indicates nearly the end of the first hour (and so is likely still usable); the dial is illuminated by the Sun for just over three quarters of the daytime. 

This demonstrates the limited usability of this type of dial when the Sun is North of the celestial equator - surely there were complementary dials for these months? Perhaps something as simple as a North-facing dial of the same type? In this case, it would need marks for the first two and final two hours only of the daytime. In the morning, the shadow cast by the gnomon would be horizontal at sunrise and would rotate until reaching a particular angle at which time the entire dial would go into shadow and the South-facing one would then be illuminated, continuing at the correct angle and hour. Later in the day, the process would be reversed as the South-facing dial went into shadow and the North-facing one became illuminated.

Wednesday, June 3, 2026

A scratch dial

Updated 11 June 26


Also known as a mass dial, although I seem to recall that strictly speaking there is a distinction. I came across this one by chance at St. Catherine's Church, Montacute, Somerset. It's a vertical sundial that surely pre-dates the Church clock of around A.D. 1400. The hole centre bottom would have held a horizontal gnomon to cast a shadow for telling the time. 

~~~

Originally, I wrote: There are 12 lines radiating from the gnomon hole for indicating the hours from 6 am to 6 pm, albeit inaccurately, as they are spaced regularly (15 degrees) and with no regard for latitude.

However, having read the section Medieval Times and Canonical Hours here on the excellent GAUDIUM SUB SOLE website, I was reminded that centuries ago, day and night were divided into 12 hours each, an hour in the day-time being longer than at night-time in the Summer, for example. 

Then I wrote: There are 12 lines, spaced at 15 degrees, radiating from the gnomon hole for indicating the 12 daytime hours from sunrise to sunset, the length of the hour depending upon the time of year. While perhaps crude in execution, it and similar dials were sufficiently accurate for the needs of the period.

However, on further reflection, this is still only partially correct! These dials show sunrise and sunset only when the Sun is fully in the Southern sky, i.e. between the Autumn and Spring equinoxes. Between the Spring and Autumn equinoxes, for part of the day, after sunrise and before sunset, the Sun is in the Northern sky and such dials are in shadow. More to follow...

~~~

The dial is unusual, being upside down, and hence unusable! Presumably the stone was re-used when the dial was no longer needed. 

More info. on this dial from GAUDIUM SUB SOLE here, along with another dial at the church that I missed!

Note - the design was already ancient when it was made, being the same type as this Ancient Egyptian dial.

Tuesday, June 2, 2026

Stereo Pairs

There's a new page on the RetroMat website with some stereo pair pictures I've taken, for cross-eyed free-viewing. 

This one, however, is for parallel free-viewing. It's easier on the eye, but by necessity has been reduced in size.

Also, note this is an example of a hyper stereo pair, which exaggerates the depth, the pictures have been taken from points of view that are more widely separated than are human eyes.



Monday, June 1, 2026

Estimating aphelion and perihelion for Mercury and Venus

In an earlier post, I estimated the orbital radii for Mercury and Venus, by taking the sine of the average greatest elongation angles for each, using data for 2014 to 2029 from here.

In fact, I've been treating the orbits as being near-circular, but recently I've been wondering how to find their true shapes (which we know from Kepler and Newton are ellipses). Specifically, can I use the extremes of the greatest elongation angles to find accurate figures for the furthest (aphelion) and closest (perihelion) approaches of these planets to the Sun?

So, here is a summary for the inner planets. I've separated greatest elongation observations by East and West. The sines of the minimum and maximum angles within each set of observations provide estimates of perihelion and aphelion respectively and as can be seen are nearly exactly the same as the true figures, with exceptions to the exact figures highlighted with an *. This surprises me, because I'd assumed that I would have to find and make an allowance for the Earth's orbital eccentricity (loosely, non-circularity) for greater accuracy (which I have not done yet), especially so in the case of Venus, because the Earth's orbit is actually the more eccentric of the two planets.

Notice how eccentric Mercury's orbit is!

Mercury

  • Greatest Elongation West
    • 50 observations
    • min. 17.9 (sine: 0.31)
    • max. 27.8 (sine: 0.47)
  • Greatest Elongation East 
    • 51 observations
    • min. 18.1 (sine: 0.31)
    • max. 27.4 (sine: 0.46*)
  • wikipedia: perihelion 0.31 AUs, aphelion 0.47 AUs

Venus

  • Greatest Elongation West
    • 10 observations
    • min: 45.8 (sine: 0.72)
    • max 47 (sine: 0.73)
  • Greatest Elongation East
    • 10 observations
    • min. 45.4 (sine: 0.71*)
    • max. 47.2 (sine: 0.73)
  • wikipedia: perihelion 0.72 AUs, aphelion 0.73 AUs

Saturday, May 30, 2026

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!

Friday, May 29, 2026

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.

Thursday, May 28, 2026

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).

Wednesday, May 27, 2026

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