Proving Kepler's Law: A Math Challenge

In summary: That cannot be the case.In summary, the conversation is discussing how to prove that the eccentricity multiplied by the directrix is equal to b^2/a in an ellipse. The participants have found equations for the eccentricity and directrix, and are trying to show that a=b^2/a. However, this would mean the shape is actually a circle, which is not the case.
  • #1
paintednails
1
0
kepler's law!

hi, does anybody know how to prove that the eccentricity multiplied by the directrix is equal to [itex] \frac {b^2}{a} [\itex]?

i found that the eccentricity of an ellipse is equal to c/a.

i also found that the directrix is equal to a/e. the way i see it, if i multiply e and d, then i get

ed = a

but how do i prove that a = b^2 / a ?

please help, thanks <3
 
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  • #2
paintednails said:
hi, does anybody know how to prove that the eccentricity multiplied by the directrix is equal to [itex] \frac {b^2}{a} [/itex]?

i found that the eccentricity of an ellipse is equal to c/a.

i also found that the directrix is equal to a/e. the way i see it, if i multiply e and d, then i get

ed = a

but how do i prove that a = b^2 / a ?

please help, thanks <3
?? What do you mean by the directrix of an ellipse? A parabola has a directrix but it is a line, not a number.
 
  • #3
I suspect we have an ellipse with center at the origin, major axis on the x-axis, minor on the y, foci at +/-c, major axis length a, minor length b, and vertical directrices at x=+/-a/e

Ifa=b2/a, then a2=b2, which means you really have a circle.
 

Related to Proving Kepler's Law: A Math Challenge

1. How did Kepler come up with his laws?

Kepler's laws were based on his observations of the movements of the planets and his mathematical calculations. He studied the works of previous astronomers, such as Copernicus and Tycho Brahe, and built upon their theories to formulate his own laws.

2. What are the three laws of Kepler?

Kepler's three laws are: 1) The planets orbit the sun in elliptical paths with the sun at one focus, 2) The line connecting a planet to the sun sweeps out equal areas in equal times, and 3) The square of a planet's orbital period is proportional to the cube of its semi-major axis.

3. How do you prove Kepler's laws mathematically?

To prove Kepler's laws mathematically, we use the principles of calculus and differential equations to derive the equations of motion for the planets. We then solve these equations to show that they align with Kepler's laws.

4. Why is it important to prove Kepler's laws?

Proving Kepler's laws mathematically provides a deeper understanding of the fundamental principles that govern the motion of celestial bodies in our solar system. It also allows us to make more accurate predictions and calculations regarding the movements of the planets.

5. Are Kepler's laws still relevant today?

Yes, Kepler's laws are still relevant and widely used today in the field of astronomy. They have been confirmed by numerous observations and experiments, and are essential in understanding the structure and dynamics of our solar system and beyond.

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