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hello

I have read your answer to my follow up question

although the focus where the Sun must be is closer to the perihelion than the aphelion and although Sunlight is perpendicular to the equator at spring Equinox whatever was the probable day 21, 22 or 13 March, I don’t know how you do it to prove that all quarters are equal.

Anyway I will go with you, but how we could explain the new problem?

You know that the two Solstices devise the orbit longitudinally into two identical halves, and do the Equinoxes devise the orbit (year) transversally into two halves not at all identical in size but equal in time, and this because the straight line matching the two Equinoxes must pass through the center of the Sun located in a point closer to perihelion than aphelion = a positive number what it was, so we can't avoid this mathematical logic division of the orbit transversally into these two unequal halves, thus quarters can never be equal.. sir.

For more details pleas take a look on:

my facebiik page:

https://www.facebook.com/astronomicalbarzaq?ref=hl

and my blog:

http://barzaqideas.blogspot.com/

sincerely

Dr Barzaq

note: if you have a personal email where we can make a better dialog

Hello,

Sorry, but I don't give out my personal email.

Let me just say that I believe the problem is you're coming at this too much from the perspective of geometry instead of the physics underlying the 2nd law.

If Kepler's 2nd law holds at every point ("equal areas swept out in equal intervals of time") we have:

r^2 (2π/T) = h

where T is the period (time interval for orbit), 'h' is a constant ('specific relative angular momentum') which is twice the rate of area description (i.e. by the radius vector, r). Thus, if the radius vector is r1, then h = 2A1, when the radius vector is r2, then h = 2A2. Thus, the "quarters being equal refers to equal areas mapped out in roughly equal period of time t.

Here's a neat animation showing how the law works for equal areas, which I suggest you study closely. Its advantage is that it conveys a dynamic picture as opposed to a static one:

http://www.physicsclassroom.com/mmedia/circmot/ksl.cfm

Note how the areas for the orbit are narrower (as a sector of the ellipse) at greater distance and wider (greater angle theta = theta2 -theta1) at lesser distance. Near perihelion obviously the planet (e.g. Earth) goes FASTER and near aphelion SLOWER, so that is the basis of the equalizing areas.

Another way to check this if you are still skeptical, is to obtain the precise dates for equinoxes, solstices for a given year (e.g. from an ephemeris) and do the detailed area computations - quarter to quarter. You ought to find the areas between the quarters are equal, as Kepler's 2nd law indicates.

I do hope this helps, especially the link given, but if not there is not much I can do to assist, so suggest maybe going to another expert.

Astronomy

I have more than forty years of experience in Astronomy, specifically solar and space physics. My specialties include the physics of solar flares, sunspots, including their effects on Earth and statistics pertaining to sunspot morphology and flare geo-effectiveness.

Astronomy: Worked at university observatory in college, doing astrographic measurements. Developed first ever astronomy curriculum for secondary schools in Caribbean. Gave workshops in astrophysics and astronomical measurements at Harry Bayley Observatory, Barbados. M.Phil. degree in Physics/Solar Physics and more than twenty years as researcher with discovery of SID flares. Developed of first ever consistent magnetic arcade model for solar flares incorporating energy dissipation and accumulation. Develop first ever loop solar flare model using double layers and incorporating cavity resonators.**Organizations**

American Astronomical Society (Solar Physics and Dynamical Astronomy divisions), American Mathematical Society, American Geophysical Union.**Publications**

Solar Physics (journal), The Journal of the Royal Astronomical Society of Canada, The Proceedings of the Meudon Solar Flare Workshop (1986), The Proceedings of the Caribbean Physics Conference (1985). Books: 'Selected Analyses in Solar Flare Plasma Dynamics', 'Physics Notes for Advanced Level'. 'Astronomy and Astrophysics: Notes, Problems and Solutions'. **Education/Credentials**

B.A. Astronomy, M. Phil. Physics **Awards and Honors**

American Astronomical Society Studentship Award (1984), Barbados Government Award for Solar Research (1980), Barbados Astronomical Society Award for Service as Journal Editor (1977-91)**Past/Present Clients**

Caribbean Examinations Council, Barbados Astronomical Society, Trinidad & Tobago Astronomical Society.