Personal takes – Page 6

Dedekind’s theorem: a draft proof

A draft written in Italian proves the existence and uniqueness of an element that separates two non-empty sets representing the field of real numbers.
This allows us to prove that the real numbers are a complete space (i.e., that every Cauchy sequence is convergent).

Note: in the case of irrational numbers instead of rational numbers, it does not work because they are not complete, i.e., it is not possible to define an element that is an extreme.
In layman’s terms, one could say that if rational numbers are removed from the real numbers to obtain irrational numbers, then ‘gaps remain’ and completeness no longer exists.
upgrade: a reconsideration of the demonstration

Brussi 2026_Dedekind theorem

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References for laymen: Archimedes’ hydrostatic principle

Archimedes of Syracuse (287-212) was perhaps one of humanity’s greatest mathematical geniuses. He is remembered for the principle of hydrostatic thrust, the screw pump, and war machines, but his most advanced studies were on conics and mathematical methods for calculating surfaces and volumes (e.g., of sphere). These are considered among the highest achievements of ancient science and, although they lack modern formal rigor, they represent the direct precursors of the concept of limits and integral calculus (independently developed by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century). See, for example, Lucio Russo, Archimedes: A Great Ancient Scientist (Carocci Editore, 2019).

One day, my young son asked me, “Why do ships that are heavy and made of iron float?” I couldn’t explain the density of water to him, but I found a good answer for a child: “Because they weigh less than the water they displace.”
This is exactly Archimedes’ principle of hydrostatic thrust, which in more complicated terms is expressed as: “A body immersed in a fluid experiences an upward thrust equal to the weight of the volume of fluid displaced.” So if this is greater than the weight of the body, it floats.

We can express Archimedes’ principle of hydrostatic thrust (force) F(A) using the following formula:
[1] F(A) = m * g = ρ(fluid) * V(displaced) * g
where m = ρ * V is the mass, g is the acceleration due to gravity, ρ represents the density, and V represents the volume of fluid displaced by the body.
For this reason, any body also ‘weighs’ according to the density of the fluid in which it is immersed. The greater the density, the greater the hydrostatic thrust and the lower the weight of the body.

As an example, if an iron nail (which, due to its shape, does not seem to be able to float like a ship by displacing a lot of fluid) were placed in a container with liquid mercury, which has a density almost double that of iron, the nail would float at the level of the weight of mercury displaced. Calculations show that it would be immersed just over half its volume (about 58%).

In this video, we can see how an iron anvil floats in mercury because the density of iron (7.87 g/cm³) is almost half that of mercury (13.55 g/cm³) (source: Wonder of science on X channel).

Another famous example of buoyancy is that of hot air balloons: they float exactly like a ship because the volume of fluid (atmospheric air) they displace weighs more (is denser) than that contained in the balloon, which is heated with a burner (and is therefore less dense).

An hot air balloon (source: freedome.it)

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References for laymen: an experiment to verify the law of falling bodies

The first observational measurements on the law of falling bodies were made by Galileo Galilei (1564-1642).
The famous law, considering the distance d traveled during the fall, is:
[1] d = g t² / 2
where g is a constant representing the local gravitational acceleration (on Earth a mean value is g = 9.8 m/s² ), and t is the time elapsed. In more easily observable terms, the fall time can be measured reversing the relationship [1] :
[2] t = √(2d/g) .
Galileo predicted (inspired by what Domingo de Soto had already conceived in 1551) that all bodies follow this law regardless of their mass, studying rolling balls (rather than dropping them from the famous Tower of Pisa). However, everyday experience tells us that light bodies fall more slowly: this is simply due to the friction of the fluid in which they are immersed (e.g., air).

A rather expensive experiment was reported to have been carried out on the Moon, when a hammer and a feather were dropped and touched the ground at the same time (astronaut David Scott on August 2, 1971).
I propose the following more economical experiment: drop a coin that is not too small, for example a 2 euro coin, flat on its face, placing a small piece of paper, e.g. 5 mm in diameter, on top of it. A good height would be half a meter. You can observe that the piece of paper reaches the ground at the same time as the coin, which prevents the air from slowing it down.
You could also place the piece of paper in a transparent container that you then drop, but this would make it less clear to observe.

As a side note, it can be observed that a body in free fall accelerates until this acceleration equals the acceleration of gravity, due to the mass of the body (tipically the Earth) on which it is falling (possibly slowed down by air resistance, known as drag force). Once this is reached, the speed of fall, known as terminal velocity, remains constant. On contrary, a body falling in a vacuum never reaches terminal velocity because there is no drag force; it accelerates indefinitely as long as it is falling.

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References for laymen: how to use the sky to find your orientation

At night, everyone knows that North is roughly indicated by the North Star, which is easy to locate using the well-known asterism of the Big Dipper (and/or the slightly less prominent Little Dipper), see Figure 1.

Figure 1

During the day, however, the easiest direction to identify is South, which is accurately indicated by the Sun at its culminate point at noon every day of the year. Without having to wait for the culmination, knowing the time, you can estimate the direction of South by adding to the direction of the Sun at that moment (in the northern hemisphere, to the right in the morning and to the left in the afternoon) the angle missing to reach noon, considering 15° for each hour to midday (a full rotation of the Earth of 360° divided by 24 hours = 15°).
See Figure 2 for some practical references for estimating angles in the sky.

The height in degrees of the Sun above the horizon at its culmination point has been used for thousands of years to determine the observer’s latitude. Eratosthenes of Cyrene (276-194) used it to calculate the circumference of the Earth in a way that was effective for the knowledge of the time.

And again, thanks to knowing the exact ‘universal’ time, you can determine the longitude by converting the hours/minutes difference between the local culmination time of the Sun and the culmination time at longitude zero, taken as a reference, at the Greenwich meridian, into degrees. Obviously, east of this meridian, the culmination will be earlier, while west of it, it will be later.
This principle has also been known for centuries. However, it was possible to exploit it only when clocks of sufficient precision and reliability were built (at the beginning of the 18th century, mainly thanks to John Harrison, see for example Longitude, Dava Sobel, Walker Publishing Company, 1995).

To obtain accurate measurements, precision instruments (like a sextant) are obviously required (apart from the GPS available in every cell phone today!), but for a rough estimate, the above is sufficient.

Figure 2 – Easy reference for measuring angles in the sky (image credit: Nightwatch, Terence Dickinson)

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References for laymen: angles in the sky

The positions and the sizes of cosmological objects observed as projected onto a background sphere are measured in degrees, starting from reference orientations. This brief note aims to highlight the observational ‘measurements’ of certain cosmological objects, measurements that are rarely taken into consideration.
We imagine a simple observation of the night sky with the naked eye…

An approximate idea of the measurements in degrees can be obtained from Figure 1. Other references are the measurements of the Sun and Moon, both approximately 0.5° (which is why we have total solar eclipses).
Since they cannot be seen ‘at a glance’ because they are not very bright, we do not realize that some cosmological objects are actually ‘large’ in the sky.

Figure 1 – Easy reference for measuring angles in the sky (image credit: Nightwatch, Terence Dickinson)

For example:
– the Andromeda galaxy (Figure 2) measures about 3° x 1° (so it is 6 times the size of the Moon)

Figure 2 – Andromeda galaxy (image credit: Westend61 via Getty Images)

– the Orion Nebula (Figure 3, the star-forming region closest to us in our galaxy, about 1350 light-years away) measures about 1° x 1° (i.e., twice the size of the Moon)

Figure 3 – Orion Nebula (image credit: NASA/ESA’s Hubble Space Telescope)

– The Large Magellanic Cloud (Figure 4, visible in the southern hemisphere) measures approximately 11° x 9° (i.e., it is 22 times wider than the Moon).

Figure 4 – Large Magellanic Cloud (image credit: Spitzer Space Telescope by NASA)

– Halley’s Comet (Figure 5), which passed by in 1986 (and will pass by again in 2061), measured a maximum of 15° (i.e., 30 times the width of the Moon).

Figure 5 – Halley’s Comet (image credit: W. Liller, Easter Island, part of the International Halley Watch IHW)

For comparison with the nearby planets in our solar system:
– Jupiter (whose diameter is about 11 times that of Earth) can reach a maximum of 50″ (or only 1/36 of the width of the Moon)
– Saturn (whose diameter is about 9.5 times that of Earth) can reach a maximum of 20″.

Credits: Teaching material for Spherical and Practical Astronomy course, Prof. Enrico Maria Corsini (University of Padua, Italy)

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Brief on Kramers’ Opacity Law in Stellar Astrophysics

In stellar astrophysics, opacity ( k ) is a measure of a material’s resistance to the flow of radiative energy. One analytical descriptions of this phenomenon is Kramers’ opacity law, a power-law relation derived from atomic physics that characterizes how stellar matter interacts with radiation under specific thermodynamic conditions.

Brussi 2026_Brief on Kramers’ Opacity Law in Stellar Astrophysics

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Chandrasekhar dynamical friction formula

Brief analysis of Chandrasekhar’s formula on dynamic friction in the general case of galactic encounters in the study of Galactic Dynamics.

Spoiler: we get a result for the dynamical friction timescale which means that in a time span of a few Gyr, for instance with the values of our assumptions applied to the Milky Way, the orbit of a massive satellite has substantially reduced, while the orbit of lower mass satellites (such as globular clusters) has not been significantly affected by dynamic friction.

Brussi 2026_Chandrasekhar dynamical friction formula

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Note for laymen – what is the dark matter?

Let’s consider a disc galaxy with billions of stars forming the disc, which rotates (the Andromeda galaxy in the photo, to give a well-known example). We can consider for each of these stars that the centripetal acceleration
[1] a’ = v²/R
is equal to the gravitational acceleration
[2] a” = G M/R²
therefore from [1] = [2] we obtain:
[3] v = √(G M/R)
where v is the local velocity of the source considered, R is its distance from the galactic center, G is the gravitational constant, M is the total mass contained within the radius R (which determines the gravitational attraction).
Measuring the rotational velocities of galaxies is relatively easy, if they are not seen face on, and it is generally observed that throughout the disk the rotational velocities of the sources (stars but also gas clouds) are almost constant. Therefore, from relation [3] it follows that the mass M must increase in proportion to the radius R , since G is a constant. Indeed, the mass contained within the radius R increases as R increases, but what is observed is not sufficient to justify the constant value of v . These considerations leads to the hypothesis that there is a substance that is invisible and non-baryonic in nature (i.e., not made up of protons and neutrons, which are not detected) that manifests itself only through gravitational behavior, which is called dark matter.

Processed photograph of M31, Andromeda. It was not possible to obtain the author of this beautiful photograph.

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Ole Rømer and the finite speed of light

The first to observe that the propagation of light was not instantaneous was the astronomer Giovanni Domenico Cassini (1625-1712); the first to estimate its speed, in 1676, was the Danish astronomer Ole Christensen Rømer (1644-1710), both based their observations on the solar system.

Brussi 2026_Ole Rømer and the finite speed of light

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