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Astronomy/Kepler 12 equal areas


According to geography, the Sunlight through the four seasons must be:
1-   Orthogonal to the tropic of Capricorn at December 21=winter Solstice
2-   Orthogonal to the tropic of cancer at Jun 21 = summer Solstice
3-   Perpendicular with the equator at March 21 = spring Equinox
4-   Perpendicular with the equator at September 21 = fall Equinox
But in the image of "the law of equal areas" illustrated at
where Kepler has devised the space of Earth's orbit into twelve equal areas in order to prove his second law, we can see the two Solstices of winter and summer wher the Sunlight is orthogonal to tropics, but at March 21 (area "C" or 3) and September 21 (area "I" or 9), the Sunlight is not perpendicular to the equator which abolish the definition of the Equinox because, in this case, dark and day are not equal.
1-   have you any comment or explanation?
2-   if you see in this image violation to geographical rules what correction you suggest?
3-   If it is impossible to redivide the space of the orbit into 12 equal areas without violating the laws of geography, can we take  this as an evidence of the collapse of Kepler second law?
4-   Would you send me a corrected image pleas?
Thank you

Note: I 1- gave the link because I didn't succeeded in uploading the image.
2- this is not a homework, I'm E N T specialist.

I'm sorry but the link you provide on the equal areas shows (to me) no such evidence that:

"the two Solstices of winter and summer where the Sunlight is orthogonal to tropics, but at March 21 (area "C" or 3) and September 21 (area "I" or 9), the Sunlight is not perpendicular to the equator which abolish the definition of the Equinox because, in this case, dark and day are not equal."

given the diagram was not of such scale as to show that, and indeed is designed to show the areas law aspect, not equinoxes, or solstices.

Your original link diagram was intended to show those.

Hence, there is NO need for me to make any corrections and Kepler's 2nd law still holds.

As I tried to explain, such conventional 'season' diagrams - as depicted in the original link you sent me - aren't drawn to scale (or shown with Sun at one focus) because the emphasis is on showing the axial tilt remaining constant through the orbit.  So one is not going to try to put the Sun at the focus or even show an elliptical orbit given that's of secondary import to the depiction of seasons.

Conversely, in the case of area digrams one is not going to try to show equinoxes, solstices. And to try to interpret them or any "geographical geometries"  from such diagrams is misdirected and useless.

Astronomical diagrams are adapted to what they have to show or intend to show. In most cases they aren't intended to be taken literally - because the scales are off.

Anyway, I see no further exchange on this can be useful. Perhaps you should contact another expert.


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Philip Stahl


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.

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

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'.

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.

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