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The two tropics: capricorn and cancer

• QUESTION: Dear astronomer, NASA has measured the exact timing when the earth would be closest to the sun, or farthest away from the sun. NASA's calculations on this regard assigned Jan 4, and on July 4 to be the time when earth would be closest and farthest away to the sun respectively. My question is this: if Jan 4 is the date over which the earth is closest to the sun, then on that same date the deceleration of the planet earth during its orbit around the sun would reach to zero, and thus the earth would start to reverse its direction from going farther away from to the sun and start to get closer to the sun due to the force of gravity of the sun on earth. At this point of time the earth start to accelerate to get nearer to the sun at the same time the momentum of the earth which is calculated to be M*V would prevent the earth to fall towards the sun and thus reversing the direction of the earth movement and causing the earth to build its acceleration day after day until it reaches maximum acceleration at the time of the equinox that supposedly falls on March 21. At this point the earth start to decelerate until it has reached the other side of the elliptical shape, the path of which the earth makes its revolution around the sun. On July 4, the earth deceleration reaches zero and the reversing earth direction to complete the other have of the ellipse by building up acceleration and start again to reach maximum acceleration during its orbit around the sun on September 21, the time of which earth attain maximum acceleration. The earth then starts to reduce once again its acceleration (Or causing the earth to decelerate) while orbiting the sun until it is on the other side of the ellipse reaching zero deceleration on January 4. If January 4, and July 4 are the points of which the earth reverses directions when its deceleration reaches zero, then at that exact location the sun must be perpendicular over the tropic of Capricorn on Jan 4, or perpendicular over the tropic of Cancer on July 4. If this is the case then why do we assign December 21, and July 21 as the time when the sun is perpendicular over the tropic of Capricorn, or tropic of Cancer? It is impossible that the sun becomes perpendicular over the two tropics except on those assigned dates, namely January 4, and July 4 every year. Can you explain how the sun is perpendicular over the two tropics of Capricorn and Cancer on the December 21, and June 21 every year notwithstanding that the zero deceleration of earth occurs only on January 4, and on July 4? Thank you! So much! ANSWER: Hello, Somehow or other, you have conflated the positions of Earth in its orbit (including changes in radius vector, and acceleration) with the PROJECTED positions of the Sun on the celestial sphere. Thus, you interpret Jan. 4 (when Earth is nearest the Sun or at perihelion) as having some connection to the winter solstice (~ Dec. 21). But what really are the Tropic of Cancer, Tropic of Capricorn designations in context? In fact, they signify the upper and lower bounds -positions on the CELESTIAL SPHERE where the Sun attains it maximum (+) and minimum (-) DECLINATIONS, respectively! One can even devise an instrument to record-measure the hour angle of the Sun daily, and plot it on a graph of declination vs. date. One will see: a) the graph has a sinusoidal form (e.g. y = sin(x)) b) the maximum occurs at (+23 deg 27') (projected back to Earth as the Tropic of Cancer) c) the mininum occurs at (-23 deg 27') or the Tropic of Capricorn projected back at Earth d) the zero positions (decl. = 0 deg) occur at Sept. 21 and Mar. 21 Thus, there is nothing whatever amiss. Further insight discloses that for six months of the year the Sun's declination is positive, and for the other six months, negative. Again, nothing amiss here. The Earth in its own right obviously undergoes a variable motion in its circuit round the Sun. The speeds vary as well as accelerations. But you should know that in tracing the APPARENT motion of the Sun across the celestial sphere, these variations are disregarded and one has a "uniform" Sun or what is often called "the mean Sun". Secondly, the distance of the Earth from the Sun has nothing to do with the location (pinpointing) of the Sun's declination. The fact therefore that the Earth is at perihelion on Jan. 4 is no use at all in trying to ascertain when the Sun attains maximum declination, or minimum. And on Dec. 21, the winter solstice, the Sun is directly over the latitude that projects onto the celestial sphere at (-23 deg 27') or the Tropic of Capricorn. This is why the Sun's altitude is lowest in the sky for a northern hemsiphere observer. It would be highest at the summer solstice (June 21) when the Sun is over the Tropic of Cancer or at a declination of (+23 deg 27'). Hopefully, this helps. ---------- FOLLOW-UP ---------- QUESTION: Thank you so much for your answer. I understand that from our observation of the sun declination, the sun reaches +23.7 degrees on July 4, and -23.7 degrees on December 4. This is accepted by all astronomers that this is true because that is how the observation of our sun is recorded day after day. Now the Earth axis of rotation around itself is also inclined 23.7 degrees from the line connecting the two poles. This tilt is a permanent tilt and does not change year after year, notwithstanding that the wabbling of the earth does not change the tilted axis, but does change the direction of the polar star over the ages. I understand that due to the wabbling effect of the earth the earth makes one whole circle in the northern and Southern sky area wherein Polaris which is now the polar star lies on that circle. Also, due to that wabbling effect, the polar star changes in a rate of one degree every 72 years. Thus the Polar star at the time the Pyramids were built was the star Thu-ban (in alpha Draconis). This star lies on that circle created by the wabbling of the earth, which completes one whole revolution every 26,000 years approximately. In addition, we know that the star Vega will become the north star in the future (I don't recall exactly when). You explained the mechanism of the elliptical revolution of the earth around the sun by observation only and not by theory or using mathematical logic. With all respect to your kind self, you could not explain the phenomenon why DEC 21 and Jan 21 are the the winter and the summer solstices. If we apply mathematical theory on this situation, considering that the earth tilt remains 23.7 degrees whether it is on the tropic of Cancer or Capricorn. I know very well that the distance of the earth from the sun has nothing to do with when the two seasons begin. But if our observation shows that this is the case, then we must justify why is it this way. Now when the earth which has a 23.7 degrees tilt from the line connecting the earth north pole with the earth south pole remains always as it is whether it is on this side of the ellipse or the other side of the ellipse, then mathematically speaking, it is impossible for the sun to be perpendicular over the two tropics except at the time the earth reverses direction, namely on Jan 4, and July 4 every year. To make my question very clear, take the point when the earth is closest to the son on January 4. At this time the 23.7 degrees tilt of the earth pointing away from the sun but the polar star remains the same whether it is January 4, or July 4. The north pole of our earth keeps the polar star fixed all year around! The sun cannot be perpendicular over the tropic of Capricorn on January 4, 15 days earlier before the earth start to reverse its direction to complete the elliptical circle around the sun. Those 15 days must make the sun perpendicular over another latitude closer to the tropic of Capricorn but not until the reverse of earth direction would the sun become perpendicular over the tropic of Capricorn. This is true because on September 21, as the astronomers tell us, the sun is exactly over the equator. Now between the equator and the tropic of Capricorn we have a quarter of the whole elliptical path of earth until the sun becomes over the tropic of Capricorn. This is when the earth begins to turn back to the other half of the ellipse. This occurs only when the earth changes direction on January 4. I understand how you explained the whole situation for me. You said: Our observation shows the the winter solstice begins on December 21, but the truth that I can derive from mathematics tells me that the earth is closest to the sun on December 21 and cannot be January 4. NASA is not explaining the phenomenon mathematically. I can't understand this mechanical situation of earth movement using mathematics. You are telling me, that how we observe the situation to be and at the same time, you emphasize that January 4 is the point when earth reverse direction. From the observation point of view, you are correct. From a mathematical point of view NASA is wrong. Unless you use mathematical principal to explain it to me! What you have just told me is that is what we observe. You yourself should question it more than me because you seem to be skeptical too about this and justify the mechanism from observation. What I am say negates what NASA is saying from a mathematical principal. The question remains open; there is something undiscovered and untold in your answer. However, I truly thank you for your time taken to answer my question, but don't you think there is something wrong in NASA's findings by showing the most extreme points of the earth around the sun are those that come on January 4, and on July 4. If you don't understand my logic, then please disregard my question. I will have to continue to look for an answer to satisfy my understanding on this regard. Until now, I say NASA is wrong if we apply mathematical theory to do the calculations. Thank you so much for the answer, God bless you! ANSWER: Hello, Plowing through your script, I think the bugbear in the works (and what seems to be troubling you) is the Earth "reversing direction" on Jan. 4 - as you put it. In fact, technically speaking, this is not the case. What is happening is that the y-component of acceleration (taking the Earth's motion in the x-y plane of the ecliptic) is going from (+) to 0 to (-) sign. There is no literal 'reversal' of orbital directon here! Let the instantaneous radial distance at any time t (r - or the radius vector) be: r = (x^2 + y^2)^1/2 then the respective x and y components of acceleration are: a(x) = dv(x)/ dt = -GMx/ r^3 a(y) = dv(y)/ dt = -GMy/ r^3 If you were to construct a planetary motion model based on the above and some initial conditions (e.g. x(0) = 0.25, y(0) = 0) you would easily see the point I am making and and how the COMPONENT a(y) direction is changing NOT the orbital direction of the Earth. I suggest what you do is try to obtain Feynman's Lectures on Physics, Volume I, and go to chapter 9, specifically Section 9-7. There he takes you through all the basics of constructing a planetary motion model with the position, velocity and acceleration components attended to. A simple computational program (following Feynman's prescription) can easily generate this model (like a Mathcad file) and generate an orbit plot like the one Feynman shows on page 9-7 with numerical display (output) that he shows on page 9-8 (Table 9-2). If you get the book, pay attention to the Table and the THIRD column from the right for component a(y) and note how it goes to 0 then negative. THIS is what you are referring to, I believe. In any case, I am confident if you work out the model illustrated (by Feynman) on your own, to obtain the with the outputs shown, you will remove the "mathematical problems" you seem to have on this - and also see there is zero impact on the Sun's maximum and minimum declinations, which is more determined by the tilt of the polar axis than any instantaneous orbital position. Feynman's Lecture series is pricey, but you ought to be able to get hold of Vol. 1 at a library. If you are very lucky, they may also have the audio replay of the lecture, which has been available for some time now. It can only help to improve you understanding of the situation. This is a further follow-up to your query. It is quite obvious to me that you are extremely confused. You wrote, for example: "I know very well that the distance of the earth from the sun has nothing to do with when the two seasons begin. But if our observation shows that this is the case, then we must justify why is it this way. Now when the earth which has a 23.7 degrees tilt from the line connecting the earth north pole with the earth south pole remains always as it is whether it is on this side of the ellipse or the other side of the ellipse, then mathematically speaking, it is impossible for the sun to be perpendicular over the two tropics except at the time the earth reverses direction, namely on Jan 4, and July 4 every year." THIS is where you are wrong. It is NOT "mathematically impossible" it is just that you do not understand the geometry. I suggest going to this site and studying the diagram carefully. Note the Earth's polar axis tilt remains fixed. This axial fix (to cause seasons) has NOTHING to do with the long term precession (gyroscopic action) over 26,000 years. (By which I mean that the fact the long term precession will alter the direction to which the pole points, does not mean that NOW the pole will change directions on either side of Earth's elliptical orbit!) http://csep10.phys.utk.edu/astr161/lect/time/seasons.html This site is very good as it also covers other misconceptions people have about Earth's tilt. Finally, "mathemtically speaking" it is NOT impossible for the for the Sun to be perpendicular over the respective tropics on Dec. 21 and June 21. (NOT Jan. 4 and July 4 which again have NOTHING to do with it!) What I suspect, and it is not clear by any means, is that you want to believe or think it consistent to believe - that the variation (in changing sign) of the Earth's acceleration component (a(y)) must somehow follow the change in sign of the Sun's declination. This is born out again when you insist: "The sun cannot be perpendicular over the tropic of Capricorn on December 21 (my correction of what you had which was "Jan. 4"), 15 days earlier before the earth start to reverse its direction to complete the elliptical circle around the sun. " But it CAN!! Again, because one (the solstice position arising from the tilt of the axis) has NOTHING to do with the variation of the acceleration in orbit! (Or the perihelion position). Just because perihelion occurs on Jan. 4 does not mean Earth must be over the Tropic of Capricorn. Here is where your reasoning goes amiss. Confirmed when you also write: " I know very well that the distance of the earth from the sun has nothing to do with when the two seasons begin. But if our observation shows that this is the case, then we must justify why is it this was" But clearly you don't know! Else you'd not be always trying to reference Jan. 4 as the solstice date rather than Dec. 21! One more thing FYI, we do not have to justify WHY this is the case, any more than we have to justify WHY the Sun shines. Science is not interested in 'why' questions but HOW questions. Here is what I also suggest you do if you want me to help you: Simplify your communication to a question (ok, two at most) of NO MORE THAN TWENTY WORDS (each)! Leave out all the descriptions and irrelevancies - because to me they merely show your thinking is cluttered. For example: How is it that the Earth's orbital acceleration component does not change with the Earth's declination? Again, you can use two questions, but in any event keep them to UNDER 20 words max. I found this also to be very effective strategy when I taught physics, and students experienced problems in asking clear and concise questions. A word limit will force you to gather your thoughts more coherently and leave out dross and immaterial stuff. If you are unable to break down the problem into no more than TWO questions of 20 words each, then it is obvious to me the problem is not even clear in your mind. When you make it clear, please feel free to get back to me. ---------- FOLLOW-UP ---------- QUESTION: Ok, I think I now got where my foolishness stems from. I am glad that my question triggered a willingness within your mind to answer my question. I humble my self before you, and I thank you for the time you have spent to correct the stubbornness in my head. I even went bizarre to accuse NASA to be incorrect while the only one who is ignorant on this issue is I myself. How stupid of me not to have visioned it in its exact mechanism. However, I have one thing against you, please accept my criticism: Had you presented the whole mechanism of earth orbit to be similar to a comet that is coming from far, far away in space to make a very fast turn around the sun converting thereby all its potential energy into kinetic energy, I would have then understand why would the winter season start 14 days earlier than January 4, and why then would the sun be perpendicular over the tropic of Capricorn. I wonder how high your IQ is! It must be beyond the 150 level. God has blessed you with an intellectual mind that I admit to be much higher than mine. Please forgive my stupidity. I thank you for using your kind heart and spending the time only to make me understand the correct function that applies for the orbit of earth around the sun. 14 days earlier from January 4, the sun would be perpendicular over the tropic of Capricorn because only then and on this exact position the sun becomes perpendicular on the tilted plane as the earth is still increasing in its velocity to reach its maximum right on January 4. At that point the sun would not be perpendicular over the tropic of Capricorn but would have started to rise above the tropic. Would thank you so much, and may God bless you abundantly be sufficient to render to you your kindness and the time you have spent for me! If I would rate you, I surely would not hesitate to assigning the highest grade for you! I also pray that you let Christ (The one who set the earth to orbit the sun) work more wonder in your brilliant mind and make Him as a brother to you who knows much , much more than you know yourself. Believe me that if I had stumbled on a very simple question like this one, there are many question in your mind that are of higher complexity you would stumble upon. If you do, then ask Him; He will answer you and would be as kind as you have been with me! God bless America!

• Answer:

  Hello, I am very glad to have been able to help you, Mazin. However, I still beg to differ that using any examples to do with a "comet" would have helped in the explanation and have been relevant to the Sun being at the zenith over the Tropic of Capricorn (on Dec. 21). As noted, this is totally a function of Earth's axial tilt remaining fixed throughout the orbit and nothing else. This was why I included the link with the diagram showing Earth in its orbit thorugh the year - something I now admit I ought to have done earlier in response. As for "stupidity" - no, that is not the issue at all. What I have seen in my experience is simply that people of all ages can get certain conceptions fixed erroneously in their head -for whatever reason- and then we try to resolve those problems. Sometimes they are relatively easy to resolve, other times not. The main thing is recognizing how and what you had misconceived, and if you have done that then I was at least moderately successful in my answer(s). All the best!