How can we derive Kepler’s third law of planetary motion, considering elliptical orbits?
History about kelper (tacho brahe)and how he started to observe
Proof thorugh eccentricity
Terminology of the formula
What does it mean
Explaining the formula through newton’s law of gravitation and motion
Finding the correlation between the second law
Mass determination (can be calculated)
Many body problems (for ‘improvement’ for the end)
Since early ages people started observing the sky, in order to have references to identify sense of time. That had led our ancestors to conclude that there are orbital motions present on daily basis. First, Claudius Ptolemy (c. 100 – 170 AD) introduced Geocentric Model where other planets and the Sun were orbiting around the Earth. By the fifteenth century, new model was introduced by Nicolaus Copernicus as the Heliocentric model, where the Sun is center of the universe and planets are circular orbiting around it. Based on these models, Tycho Brahe (17th century) made observations in order to synchronize them. He spend over two decades on observations on positions of stars and planets, where in the year 1600, Johannes Kepler joined him as an assistant. T. Brahe died without drawing a conclusion to his research, leaving Kepler with the data to finish it.
Johannes Kepler was German astronomer who is famous for his work of three laws of planetary motion.
First law, which states that all planets are orbiting in shape of elliptical orbits rather than circular around the Sun, was revolutionary both for Physics, Mathematics and Astronomy. As well, he had shown that planets are moving with a non-uniform velocity, which satisfies his second law or the Area Low. Second law indicates that lines who are joining the Sun and planets, gives equal areas in equal times. Third law states that cube of planet’s distance from the Sun is proportional to square of its period.
After his death, based on his work, Isaac Newton formulated the Law of universal Gravitation, the Law of motion and differential calculus. Even though Kepler’s laws were descriptive (explaining how the planets move), they weren’t explanatory (why they move). Therefore, I. Newton succeeded in deriving Kepler’s laws. Moreover, by using his law of mechanics (inverse square law force), he justified the proportionality constant in Kepler’s third law.
INTRODUCTION INTO ELLIPSES
In order to understand and to derive Kepler’s third law, it is essential to understand the nature of the ellipse (referring to the First law). An ellipse is the set of all points on the plane whose distance from two focal points add up to a constant. An ellipse (see Fig.1.0) is defined by set of points, which gives the equation
where a is a constant, presenting semimajor axis (half of the length of the long axis of the ellipse) and r and R represent the distance to the ellipse form two focal points (the foci), A and B.
976630107485Figure 1.0 the Geometry of an elliptical orbit
Figure 1.0 the Geometry of an elliptical orbit
If there is a situation where A and B are located on the same point, then r = R and the equation (1.0) would reduce to r = R = a, representing an equation for a circle. Therefore, it can be concluded that a circle is a special case of an ellipse. The distance b is representing semiminor axis.
In order to continue and to approach a derivation for Third law, there is a concept from the proof of Kepler’s first law that is needed. That is eccentricity. The distance of either focal points can be expressed as ae, where e is defined as eccentricity of the ellipse (0?e<1). Eccentricity shows how much a conic section (a section/slice through a cone) varies from being circular. It can be defined as the ratio of the distance between the foci (2ae) to the major axis (2a) of the ellipse. For a circle, e=0.
To find a suitable relationship with a, b and e, one of two points needs to be on semiminor axis which will form r=R. From the equation (1.0) this case is r=a, and by Pythagorean theorem, r2=b2+(ae)2. Substituting equation (1.0) into it, leads to an expression
It is more suitable to express planet’s orbit in polar form indicating its distance r from the principal focus (main focus, while other focus is an empty space) B, in terms of an angle ? (measured clockwise from the major axis of the ellipse Fig 1.0). By using the Pythagorean Theorem and trigonometric functions,
R2=(rsin?)2+(2ae+rcos?)2which by expansions and use of trigonometric identities, reduces to
R2 = 4aeae+rcos?+r2. (1.2)
Using the equation of definition of an ellipse (1.0), and substituting into (1.2) instead of R, it can be deduced that
Using Eq. (1.3), in real world example, it is possible to define the variation in distance of a planet from the principal focus throughout its orbit. Taken for example Mars, whose orbit is 2.2794×1013cm (1.5237 AU) and the Mars’s orbital eccentricity is 0.0934. When it is closest to the Sun ?=0°. Using the formula, its distance from the Sun, as a principal focus, is given by
rm= a1-e21+e=a1-e =1.3814 ??
Kepler’s third law states, that the square of the elliptical orbital period (T) is proportional to the cube of the semimajor axis (a). Where k is a constant, known as the Kepler’s constant, which is the same for all bodies orbiting the same star (focal point). In the Solar system, k has a value approximately equal to 1 year2AU-3.
In order to induce Kepler’s third law, period T needs to appear in the formulas.