- Kepler's 3 Laws
- The law of universal gravitation and Kepler's third law
- Solution b
- Experiment
- materials
- Process
- Calculation of the area of the elliptical section
- Verification of the law of equal areas
- References
The Kepler 's laws of planetary motion were made by the German astronomer Johannes Kepler (1571-1630). Kepler deduced them based on the work of his teacher the Danish astronomer Tycho Brahe (1546-1601).
Brahe carefully compiled the data of planetary movements over more than 20 years, with surprising precision and accuracy, considering that at the time the telescope had not yet been invented. The validity of your data remains valid even today.
Figure 1. The orbits of the planets according to Kepler's laws. Source: Wikimedia Commons. Willow / CC BY (https://creativecommons.org/licenses/by/3.0)
Kepler's 3 Laws
Kepler's laws state:
-First law: all planets describe elliptical orbits with the Sun in one of the foci.
This means that the ratio T 2 / r 3 is the same for all planets, which makes it possible to calculate the orbital radius, if the orbital period is known.
When T is expressed in years and r in astronomical units AU *, the constant of proportionality is k = 1:
* An astronomical unit equals 150 million kilometers, which is the average distance between the Earth and the Sun. The Earth's orbital period is 1 year.
The law of universal gravitation and Kepler's third law
The universal law of gravitation states that the magnitude of the gravitational force of attraction between two objects of masses M and m respectively, whose centers are separated by a distance r, is given by:
G is the universal constant of gravitation and its value is G = 6.674 x 10 -11 Nm 2 / kg 2.
Now, the orbits of the planets are elliptical with a very small eccentricity.
This means that the orbit is not very far from a circumference, except in some cases such as the dwarf planet Pluto. If we approximate the orbits to the circular shape, the acceleration of the planet's motion is:
Since F = ma, we have:
Here v is the linear speed of the planet around the Sun, assumed static and of mass M, while that of the planet is m. So:
This explains that the planets farther from the Sun have a lower orbital speed, since this depends on 1 / √r.
Since the distance the planet travels is approximately the length of the circumference: L = 2πr and it takes a time equal to T, the orbital period, we obtain:
Equating both expressions for v gives a valid expression for T 2, the square of the orbital period:
And this is precisely Kepler's third law, since in this expression the parenthesis 4π 2 / GM is constant, therefore T 2 is proportional to the distance r cubed.
The definitive equation for the orbital period is obtained by taking the square root:
Figure 3. Aphelion and perihelion. Source: Wikimedia Commons. Pearson Scott Foresman / Public domain
Therefore, we substitute r for a in Kepler's third law, which results for Halley in:
Solution b
a = ½ (Perihelion + Aphelion)
Experiment
Analyzing the motion of the planets requires weeks, months, and even years of careful observation and recording. But in the laboratory a very simple experiment can be carried out on a very simple scale to prove that Kepler's law of equal areas holds.
This requires a physical system in which the force that governs the movement is central, a sufficient condition for the law of areas to be fulfilled. Such a system consists of a mass tied to a long rope, with the other end of the thread fixed to a support.
The mass is moved a small angle from its equilibrium position and given a slight impulse, so that it executes an oval (almost elliptical) movement in the horizontal plane, as if it were a planet around the Sun.
On the curve described by the pendulum, we can prove that it sweeps equal areas in equal times, if:
-We consider vector radii that go from the center of attraction (initial point of equilibrium) to the position of the mass.
-And we sweep between two consecutive instants of equal duration, in two different areas of the movement.
The longer the pendulum string and the smaller the angle away from the vertical, the net restoring force will be more horizontal, and the simulation resembles the case of movement with central force in a plane.
Then the described oval approaches an ellipse, such as the one that planets travel.
materials
-Inextensible thread
-1 mass or metal ball painted white that acts as a pendulum bob
-Ruler
-Conveyor
-Photographic camera with automatic strobe disk
-Supports
-Two lighting sources
-A sheet of black paper or cardboard
Process
Assembling the figure is needed to take photos of multiple flashes of the pendulum as it follows its path. For this you have to put the camera just above the pendulum and the automatic strobe disk in front of the lens.
Figure 4. Assembling the pendulum to check that it sweeps equal areas in equal times. Source: PSSC Laboratory Guide.
In this way, images are obtained at regular time intervals of the pendulum, for example every 0.1 or every 0.2 seconds, which allows us to know the time it took to move from one point to another.
You also have to illuminate the mass of the pendulum properly, placing the lights on both sides. The lentil should be painted white to improve the contrast on the background, which consists of a black paper spread on the ground.
Now you have to check that the pendulum sweeps equal areas in equal times. To do this, a time interval is chosen and the points occupied by the pendulum in that interval are marked on the paper.
A line is drawn on the image from the center of the oval to these points and thus we will have the first of the areas swept by the pendulum, which is approximately an elliptical sector like the one shown below:
Figure 5. Area of an elliptical sector. Source: F. Zapata.
Calculation of the area of the elliptical section
With the protractor, the angles θ o and θ 1 are measured, and this formula is used to find S, the area of the elliptical sector:
With F (θ) given by:
Note that a and b are the major and minor semi-axes respectively. The reader only has to worry about carefully measuring the semi-axes and the angles, since there are calculators online to evaluate this expression easily.
However, if you insist on doing the calculation by hand, remember that the angle θ is measured in degrees, but when entering the data into the calculator, the values must be expressed in radians.
Then you have to mark another pair of points in which the pendulum has inverted the same time interval, and draw the corresponding area, calculating its value with the same procedure.
Verification of the law of equal areas
Finally, it remains to verify that the law of areas is fulfilled, that is, that equal areas are swept in equal times.
Are the results deviating a bit from what was expected? It must always be borne in mind that all measurements are accompanied by their respective experimental error.
References
- Keisan Online Calculator. Area of an elliptical sector calculator. Recovered from: keisan.casio.com.
- Openstax. Kepler's Law of Planetary Motion. Recovered from: openstax.org.
- PSSC. Laboratory Physics. Editorial Reverté. Recovered from: books.google.co.
- Palen, S. 2002. Astronomy. Schaum Series. McGraw Hill.
- Pérez R. Simple system with central force. Recovered from: francesphysics.blogspot.com
- Stern, D. Kepler's three laws of planetary motion. Recovered from: phy6.org.