Trivia 1

The states of matter depend on how the average distance between their molecules is compared with the size of the molecules themselves. In a gas, the molecules are separated in average by distances much larger than the molecules sizes. There are, as we know, substances in solid state at normal temperature and pressure. Their molecules are not individual isolated units like in the gaseous state. The separation of the molecules in a solid state can be considered in the order of the size of the molecule itself, and the force that keep the molecules together are of the same magnitude as the strength of the forces that keep the atom in the molecule together. This is typical os solids that present a crystal lattice formation.

Mars dichotomy of the hemispheres

Mars dichotomy of the hemispheres.

The red color of the surface of Mars is attributed to an unusual amount of iron oxide, . No other planet is this been seen.  This speaks of a formation that is not conforming to Bórea, which means that it has a different origin as an asteroid from the Asteroid Belt. Mars shows a dichotomy in the crust: a division between the northern and southern plains a dense formation of craters. In the south there is a flat area (Hellas) that has the same average height as the north. The clear separation of these two zones may be indicative of slanting impact with Bórea, which received the impact material that was concentrated on one of the hemispheres (south) and not a severe impact that had fused with the Earth, as hypothesized by Theia. The southern hemisphere is raised by 1-4 km above the nominal level of the Martian geoid, while the northern hemisphere is relatively mild and below this level. In Martian geology, the division between these two hemispheres refers to the  dichotomy of the cortex,  characterized by prominent geological steeps. The low gravity of Mars makes it possible to lift the mountains and volcanoes (Mount Olympus-600 km of base and 18 km high, with respect to the surrounding plains, which makes it about 27 km higher than the northern plains and the largest volcano in the Solar System).  The analysis of the meteorite ALH84001, indicates that the Martian’s soil conditions in its earlier formation, were like Earth at that earlier time of formation.   

If the proposal related to the Borea body, which evolved within the Asteroid Belt, traveling in it in a spiral trajectory  before leaving it after the accretion of asteroids at its path, is correct, and with the consequent collision of the asteroid Mars, in a mild slant tangential collision, the dichotomy of the mars crust could be explained. A collision of this nature, left Mars with a good amount of ice and other “terrestrial” material that could only sit on one half of the planet, which we now see as the southern hemisphere, leaving the northern hemisphere with occasional traces of a slight bombardment as a consequence. This collision may account for the orbit inclination of Mars of 1.8 degrees with the plane of the ecliptic. On the edge of the south-north division, you can see the action of this collision, even leaving “channels” that would leave glacials cutting these in the terrain. The many rock formations due to the action of water/ice, similar to those found on Earth, corroborate a tangential collision. This recalls the skills we try to demonstrate by throwing a stone tangentially onto the surface of the water in a river or lake, making it “jump” over the surface. If we take a film with many frames (slow motion), we could see the water stick (or dampen) only a part of the stone, leaving of course, signs of the water on the back that is dry. An analogy with what could have happened on Mars.

Lesson 16

Radio de Hill

Objects in the solar disc grow by accretion increasing their gravitational influence over the surrounding material. The Hill radios will define the distance around which the planet, in growing or full grown, will capture any object that is inside this distance. In the video I develop the notes for this distance.

The video is in Spanish language, but can be follow by English speaking viewers as well.


This second short video is to remove a possible misunderstanding in the use of the radios of the Sun in two different contexts. Sorry for the possible confusion; I thought better to publish this second video, instead of taping a new one.


Lecture 15

Las Fuerzas de Marea

Las Fuerzas de Marea son fuerzas que se ejercen sobre cuerpos (lunas, por ejemplo, o planetas) que son responsables de la deformación física de estos, debido a la acción de la gravedad. El achatamiento de los planetas es una consecuencia de las fuerzas de marea, como lo es también la fuerza centrífuga. En esta lección calcularemos el límite de Roche, para determinar la máxima distancia que debe mantener una luna, o un planeta del cuerpo alrededor del cual se traslada para evitar ser deformado severamente o desintegrado. Veremos cómo la desigualdad de Roche, puede explicar la desintegración del material de sus anillos y mantener intacta la formación de sus lunas.

Observa el video de esta lección a continuación. Por favor envíame un mensaje si te ha servido esta lección para aclarar tus ideas. Gracias.


Lecture 13

Centripetal and Centrifugal Forces

Spanish version at the end available

As indicated by the expression “centri”, while the ‘centripetal’ is directed towards the center, the ‘centrifugal is directed outwards, in the opposite direction of the center. Already with this definition there is something that does not sound complete. First, this suggests that this is the description of a circular motion. The mass in question has to be subjected to a movement around a center, such as when holding a stone attached to a string spinning around our head, when a visitor to the amusement park is entertained sitting in a swivel chair or on a rotating platform, or simply by experiencing a ‘strange’ force that tends to pull us out of the chair outward when we give a quick curve in the car we drive. Second, this force suggests that it is perceived by an observer in a non-inertial frame of reference, in which, of course, the observer is part of that system, since an observer outside the non-inertial system does not see or experience this force. Typically these forces are considered ‘fictitious’ as they disappear when there is no angular velocity, and do not follow the laws of Newton (laws of mechanics). If in a resting system S, a force F is leading to an acceleration, then according to Newton, the force can be given in the form F=ma. Bold lettering is indicating the vector nature of the physical quantity. In a rotating system S’, the acceleration a’ experienced by a body is not equal to the acceleration of the system at rest, (a a’), since the S’ system is not an inertial system. For the observer sitting on a rotating platform with angular velocity w, a body that slides (or rolls) out of the center of rotation with a velocity v’ follows a curved trajectory, created by a force perpendicular to the velocity v’ (the velocity is perpendicular to this force). The acceleration seen by the observer in S’, is given by -2 (w x v’ ) . Multiplying by the mass of the body that slides (or rolls), we obtain a force, which is not Newtonian, but a fictional force that is called the force of Coriolis:

Fc = -2m(w x v’). The observer in the rotating S’ system senses a force throwing it outwards; this force is the so-called Centrifugal Force given by the expression Fz = -mw x (w x r). Note that these two forces disappear when the platform stops rotating, or in other words when w = 0. Both forces are perpendicular to w. These forces, although considered fictional in physics, are real for an observer in the rotation system. In astrophysics, considering that the planets revolve around their axis (as well as other celestial bodies), we speak of centripetal and centrifugal forces, because, although we are making observations from an inertial system  (the Earth is an inertial system if considered fixed with respect to stars), we can consider ourselves momentarily observant in the rotational system of another planet. That allows us to talk about centrifugal forces, in case of tidal forces, for example. Once the centrifugal force disappears, the body will follow a straight-line tangent to the curve trajectory, according to the first axiom of Newton.  



Spanish version

Dos Fuerzas: La Centrípeta y la Centrífuga

Como lo indica la expresión “centri”, mientras que la ‘centrípeta’ está dirigida hacia el centro, la ‘centrífuga está dirigida hacia afuera, en dirección opuesta del centro. Ya con esta definición hay algo que no suena bien. Primero, esto sugiere que se trata de la descripción de un movimiento circular. La masa en cuestión tiene que estar sometida a un movimiento alrededor de un centro, como cuando se sostiene una piedra sujeta a un hilo girando alrededor de nuestra mano, cuando un visitante al parque de diversiones se entretiene sentado en una silla giratoria o sobre una plataforma giratoria, o sencillamente al experimentar una fuerza ‘extraña’ que tiende a sacarnos de la silla hacia ‘afuera’ cuando damos una curva rápida en el auto que manejamos. Segundo, esta fuerza sugiere que se percibe por un observador en un marco de referencia no inercial, en el cual por supuesto, el observador es parte de ese sistema, ya que un observador fuera del sistema no inercial, no ve ni experimenta esta fuerza. Típicamente estas fuerzas se consideran ‘ficticias’ ya que desaparecen cuando no hay velocidad angular, y no siguen las leyes de Newton (leyes de la mecánica). Si en un sistema en reposo S, obra una fuerza F que conduce a una aceleración a, entonces de acuerdo con Newton, la fuerza puede darse en la forma F=ma. La letra en negrilla está indicando la naturaleza vectorial de la cantidad física. En un sistema en rotación S’, la aceleración experimentada por un cuerpo a’ no es igual a la aceleración a del sistema en reposo, (a ≠ a’), ya que el sistema S’ no es un sistema inercial. Para el observador sentado en una plataforma rotatoria con velocidad angular w, un cuerpo que se desliza (o rueda) hacia afuera del centro de rotación con una velocidad v’, sigue una trayectoria curvilínea, creada por una fuerza perpendicular a la velocidad v’ (la velocidad es perpendicular a esta fuerza). La aceleración vista por el observador en S’, es dada por -m(w x v’). Multiplicando por la masa del cuerpo que se desliza (o rueda), obtenemos una fuerza, que no es newtoniana, sino una fuerza ficticia que recibe el nombre de fuerza de Coriolis: Fc = -2m(w x v’). El observador en el sistema S’ en rotación, percibe una fuerza que lo lanza hacia afuera; esta fuerza es la llamada Fuerza Centrífuga dada por la expression Fz = -mw x (w x r). Note que estas dos fuerzas desaparecen cuando la plataforma deja de rotar, o en otras palabras cuando w = 0. Note que ambas fuerzas son perpendiculars a la velocidad angular w. Estas fuerzas, aunque llamadas ficticias en la física, son reales para un observador en el sistema de rotación. En astrofísica, considerando que los planetas giran alrededor de su eje (así como otros cuerpos celestes), se habla de fuerzas centrípetas y centrífugas, ya que, aunque estamos haciendo observaciones desde un sistema inercial (la Tierra es inercial si se considera fija con respecto a las estrellas), podemos considerarnos momentáneamente observadores en el sistema rotacional de otro planeta. Eso nos permite hablar de fuerzas centrífugas, en caso de las fuerzas de marea, por ejemplo. Una vez que la rotación cese de repente, el cuerpo no será lanzado radialmente hacia afuera, sino que seguirá una trayectoria rectilínea tangencial a la curva, de acuerdo con el primer axioma de Newton.



Lecture 14

Calculating “g”

In this lecture we will explore the nature of “g”, the acceleration of gravity, from the characteristics of the centrifugal force that a body experiences on a rotating system. Besides of the fact that the attraction of bodies is described and governed by the Universal Gravitational Law, and the weight of a body is determined by that attraction, the acceleration of gravity is also influenced by the centrifugal force that is determined by the location (latitude) on the planet. The weight we have, depends on our latitude but, maybe more on how much and what we eat. If the planets would not rotate, they would not have a closed spherical form, the climate would be unpredictable irregular, life would have had a very difficult time to evolve and probably never would have done so.

Watch the lecture in the following vide.

Leave your comments.


Trivia 13

Stars That do not Set

Remember you cannot see below your horizon. Consider yourself exactly at the North Pole. You will see all the stars in circular trajectories around Polaris, that you can see as your Zenith is on Polaris. If you are exactly in the Equator, all stars are rising in the East and setting in the West. But if you are located somewhere in a latitude in between the North Pole and the Equator, you will see some of the stars rising and setting, and some stars that never will set for your latitude.

The following video, will help to understand how your latitude on Earth determines what stars you can observe that do not rise and/or set.


Trivia 12

How do we know the Earth Rotates?

Sometimes we think to have a good argument to teach that the Earth rotates by just looking at the sky and watch the stars moving around “us” from East to West, to conclude that the Earth rotates from West to East. That is all right, but it in not a prove that the Earth rotates, it is the consequence of it. The people in the Middle Ages also saw the sky moving from East to West around them, but did not conclude from this observation that the Earth rotated around its axis (what was an axis of rotation anyway?) but thought that the universe rotated around them, announcing and proclaiming that we were the center of the universe.  In modern scientific thinking, we would take that apparent motion of the sky to hypothesize a possible rotation, but have to get the information by other means. There is a famous experiment made by Jean Bernard Léon Foucault (1819-1868). He hang on a long string (67 m) a heavy mass (28 Kg) creating a pendulum from the ceiling of the Pantheon in Paris, and with it, he was able to show how the plane of the swing motion of the pendulum was changing direction, that can only be explain if the Earth rotates. Why? Because the forces acting on the pendulum (the Tension on the string) and the force of gravity (its weight) are in the same plane; no lateral forces were acting on the mass. But because the pendulum has in motion (periodic motion) and due to the angular velocity of the rotating Earth, a fictitious force, called the Coriolis force, acts on the moving mass, making the plane of the pendulum to rotate. Illustration of this phenomenon can be seen in many science museums around the world.  The Coriolis effect can be seen on large masses that move on another body in rotation, like on Earth, and the moving masses of air determining the weather patterns. You might be wondering about the word “fictitious” force. Because, it is and effect of rotation and not a real acting force. Another fictitious force is the “centrifugal” force, that is, another effect of rotation, that in many cases are being interpreted at a reaction force of the centripetal force. Those are not action-reaction pair of forces. The Foucault pendulum is the prove that the Earth rotates (and it must do it around an axis of rotation) and now, we can use this knowledge to explain the motion of the stars around us.     

¿Cómo sabemos que la Tierra rota alrededor de su eje?

A veces pensamos que tenemos un buen argumento para enseñar que la Tierra gira entorno a sí misma alrededor de un eje, porque las estrellas se mueven de Este a Oeste, en la bóveda celeste. Esto es correcto, pero no es prueba de la rotación de la Tierra, es una consecuencia de la misma. Nuestros antepasados en la Edad Media, también observaron a las estrellas moverse de Este a Oeste, pero no concluyeron que su observación estaba basada en una rotación de la Tierra sobre su eje (¿qué significaba un eje de rotación?) pero pensaban que el universo giraba alrededor de ellos, anunciando y proclamando que eran el centro del universo. En la moderna forma del pensamiento científico, tomaríamos ese movimiento aparente del cielo para hipotetizar una posible rotación, pero que la información debería de producirse por otros medios. Hay un experimento famoso hecho por Jean Bernard León Foucault (1819-1868). Colgó de una larga cuerda (67m) una pesada masa (28 kg) creando un péndulo del cielo raso del Panteón en Paris. Con este péndulo le fue posible demostrar cómo el plano de la oscilación del péndulo, cambiaba de dirección, que solamente puede explicarse por la rotación de la Tierra. ¿Por qué? Porque las fuerzas que actúan sobre el péndulo (la tensión en la cuerda, y la fuerza de gravedad: su peso) están en el mismo plano de oscilación. Pero como el péndulo tiene un movimiento periódico, y debido a la velocidad angular de la Tierra, una fuerza ficticia, llamada la fuerza de Coriolis, actúa sobre el movimiento de las masas que están sobre el cuerpo que rota, como sobre la Tierra las masas de aire que determinan el estado del tiempo. Puede que te asombres del uso de la denotación de fuerza “ficticia”. La razón es que se trata de un efecto de la rotación y no de una fuerza real actuando sobre el cuerpo, como la gravedad u otra fuerza de acción directa. Otra fuerza ficticia es la llamada fuerza centrífuga. Centrípeta significa “hacia el centro”; centrífuga denotaría “hacia afuera”, es decir, contrario de una fuerza central. Una fuerza centrípeta es dada por la fuerza de gravedad, la centrífuga es el efecto de la rotación, que puede expresarse en términos de la velocidad angular para un observador que rota con el sistema. A veces, equivocadamente la fuerza centrípeta y centrífuga, se interpretan como fuerzas de acción y reacción. No son fuerzas de acción-reacción.  El péndulo de Foucault, puede verse en muchos museos de ciencias alrededor del mundo, y es la prueba de que la Tierra rota (alrededor de un, “su” eje), y con este conocimiento, podemos explicar el movimiento de los astros alrededor nuestro.  

Trivia 11

Ursa Major – Osa Mayor

The names of the Big Dipper stars are indicated in the picture by the arrows. Their respective distances from Earth are indicated in light years (ly), and the HIP Catalog numbers. You may use Stellarium for example, and in the search window write the HIP number, and the star will be pointed to you in the constellation. The Big Dipper is the asterism of the constellation called Ursa Major. Connecting visually the stars Mizar and Alkaid, and the end of the tail of the Bear, they will point towards the star Polaris. It is a good and easy way to locate Polaris.