AREOLAR VELOCITY: HOW IT IS CALCULATED AND EXERCISES SOLVED - PHYSICAL - 2024

The areolar velocity is the area swept per unit time and is constant. It is specific to each planet and arises from the description of Kepler's second law in mathematical form. In this article we will explain what it is and how it is calculated.

The boom that represents the discovery of planets outside the solar system has reactivated interest in planetary motion. Nothing makes us believe that these exo-planets follow laws other than those already known and valid in the solar system: Kepler's laws.

Johannes Kepler was the astronomer who, without the help of the telescope and using the observations of his mentor Tycho Brahe, created a mathematical model that describes the movement of the planets around the Sun.

He left this model embodied in the three laws that bear his name and that are still as valid today as in 1609, when he established the first two and in 1618, the date on which he enunciated the third.

Kepler's Laws

In today's parlance, Kepler's three laws read like this:

1. The orbits of all the planets are elliptical and the Sun is in one focus.

2. The position vector from the Sun to a planet sweeps across equal areas in equal times.

3. The square of the orbital period of a planet is proportional to the cube of the semi-major axis of the ellipse described.

A planet will have a linear speed, just like any known moving object. And there is still more: when writing Kepler's second law in mathematical form, a new concept arises called areolar velocity, typical of each planet.

Why do the planets move elliptically around the Sun?

The Earth and the other planets move around the Sun thanks to the fact that it exerts a force on them: the gravitational attraction. The same happens with any other star and the planets that make up its system, if it has them.

This is a force of the type known as a central force. Weight is a central force that everyone is familiar with. The object that exerts the central force, be it the Sun or a distant star, attracts the planets towards its center and they move in a closed curve.

In principle, this curve can be approximated as a circumference, as did Nicolás Copernicus, a Polish astronomer who created the heliocentric theory.

The responsible force is the gravitational attraction. This force depends directly on the masses of the star and the planet in question and is inversely proportional to the square of the distance that separates them.

The problem is not so easy, because in a solar system, all the elements interact in this way, adding complexity to the matter. Furthermore, they are not particles, since stars and planets have measurable size.

For this reason, the central point of the orbit or circuit traveled by the planets is not exactly centered on the star, but at a point known as the center of gravity of the sun-planet system.

The resulting orbit is elliptical. The following image shows it, taking the Earth and the Sun as an example:

Figure 1. The orbit of the Earth is elliptical, with the Sun located in one of the foci. When the Earth and the Sun are at their maximum distance, the Earth is said to be in aphelion. And if the distance is minimal then we speak of perihelion.

The aphelion is the farthest position on Earth from the Sun, while the perihelion is the closest point. The ellipse can be more or less flattened, depending on the characteristics of the star-planet system.

The aphelion and perihelion values ​​vary annually, as the other planets cause disturbances. For other planets, these positions are called apoaster and periaster respectively.

The magnitude of the linear velocity of a planet is not constant

Kepler discovered that when a planet orbits the Sun, during its motion it sweeps out equal areas in equal times. Figure 2 graphically shows the meaning of this:

Figure 2. The position vector of a planet with respect to the Sun is r. When the planet describes its orbit it travels an arc of ellipse Δs in a time Δt.

Mathematically, the fact that A ₁ is equal to A ₂ is expressed like this:

The arcs traveled Δs are small, so that each area can approximate that of a triangle:

Since Δs = v Δ t, where v is the linear velocity of the planet at a given point, by substituting we have:

And since the time interval Δt is the same, we obtain:

Since r ₂ > r ₁, then v ₁ > v ₂, in other words, the linear velocity of a planet is not constant. In fact, the Earth goes faster when it is in perihelion than when it is in aphelion.

Therefore the linear speed of the Earth or of any planet around the Sun is not a magnitude that serves to characterize the movement of said planet.

Areolar velocity

With the following example we will show how to calculate the areolar velocity when some parameters of planetary motion are known:

Exercise

An exo-planet moves around its sun following an elliptical orbit, according to Kepler's laws. When it is at the periaster, its radius vector is r ₁ = 4 · 10 ⁷ km, and when it is at the apoaster it is r ₂ = 15 · 10 ⁷ km. The linear velocity at its periaster is v ₁ = 1000 km / s.

Calculate:

A) The magnitude of the velocity at the apoastro.

B) The areolar velocity of the exo-planet.

C) The length of the semi-major axis of the ellipse.

Answer to)

The equation is used:

in which numerical values ​​are substituted.

Each term is identified as follows:

v ₁ = velocity in apoastro; v ₂ = velocity at the periaster; r ₁ = distance from the apoaster, r ₂ = distance from the periaster.

With these values ​​you get:

Answer B)

1. Serway, R., Jewett, J. (2008). Physics for Science and Engineering. Volume 1. Mexico. Cengage Learning Editors. 367-372.
2. Stern, D. (2005). Kepler's Three Laws of Planetary Motion. Recovered from pwg.gsfc.nasa.gov
3. Note: the proposed exercise was taken and modified from the following text in a McGrawHill book. Unfortunately it is an isolated chapter in pdf format, without the title or the author: mheducation.es/bcv/guide/capitulo/844817027X.pdf