- #1
frostysh
- 63
- 3
I have a lot of questions as usually, but must begin with the difficult to understand moment which have started my explorations of the Kepler's second law. In the book of Steven Wainberg "To Explain The World", in the technical paragraph twenty one I have found the next formula which represents somekind of a small area on plane$$\large \delta A = -\dfrac{1}{2} \times r_{+} \times 2\pi r_{+} \times \dfrac{\delta\theta}{360^{\circ}} = -\dfrac{1}{2R}r_{+}^{2} \delta\theta = -\dfrac{a^{2}}{2R}\left(\dfrac{1 - e^{2}}{1 - e\cos{\theta}}\right)^{2} \delta\theta,$$where ##\delta A## is an area which has cutting the segment from the Sun to the planet on it's elliptic orbit, ##e## — is an eccentricity of the orbit, ##R## is a coeficient, the result because the proportion is in degrees but not radians, and equal ##R = 360^{\circ}/2\pi##. I think I will better just draw some picture, illustrations is very helpful indeed.
The Sun is in focus ##+c##, the planet is in point ##M## with coordinates ##\left(x, y\right)##.
My questions is: before the formula author of the book saying that it is an area ##\delta A## of the triangle which is equilateral, and it's base is on circle with radius ##r_{+}##, but is this base are approximation of the ellipse, or the circle just intersect ellipse in the point ##M##? Or this is no matters relative to the infinitely small values? I thanks for the future answers.
The Sun is in focus ##+c##, the planet is in point ##M## with coordinates ##\left(x, y\right)##.
My questions is: before the formula author of the book saying that it is an area ##\delta A## of the triangle which is equilateral, and it's base is on circle with radius ##r_{+}##, but is this base are approximation of the ellipse, or the circle just intersect ellipse in the point ##M##? Or this is no matters relative to the infinitely small values? I thanks for the future answers.