Why do the planets travel in elliptical orbit around the Sun?
Trajectories that result from inverse square central force must follow a conic section. Circular orbits are but one kind of conic section. Other conic sections are ellipses, parabolas, and hyperbolas. Planets do not have circular orbits; they have elliptical orbits. No surprise here: a perfectly circular orbit is a statistical impossibility.
If you want the mathematics behind this you will need to read a book or take a class. The subject of central force motion is covered in every sophomore/junior level classical mechanics class.
Why does the south pole (Magnetic Positive) tilt towards the Sun at the periapsis (when the Earth is closest to the Sun) and away from the Sun at the apiapsis (when Earth is furthest from the Sun). Is this gravitational or magnetic forces?
The Earth just happens to be spinning around an axis which isn't perpendicular to the plane of the orbit and which is approximately fixed in space. There are no forces involved. In the long term, this axis precesses and wobbles relative to the fixed stars. At the moment, it just happens that the time of year when the south pole is tilted towards the sun happens to be around the same time of year that the Earth is closest to the sun.
Then tell me the pattern of the rotational tilt if it is not due to the gravitational or magnetic forces.
The earth is spinning about its axis (presumably from when it was created, possibly modified by some later encounter with another body), so by conservation of angular momentum (and lack of friction in space) it just keeps spinning about the same axis, which continues to point in the same direction in space. No forces are needed. It also orbits round the sun (because of gravity), and the resulting rotation about the sun happens to be about a different axis from the spin, but the two motions are independent.
There are some slight influences on the Earth's spin from the Moon's orbit (which is in yet another different plane) and the influence of both the Moon and the sun on the tides, but those are very tiny effects.
Yes I understand that the spin has only minor influences from sun and gravitation. What influences the tilt of the planet?
You mean on a month to month basis? Nothing. The tilt is essentially constant.
I'm not sure what the confusion is here, but it may be the rotating reference frame confusion people sometimes have. When viewing things in a rotating reference frame, other objects may appear to be rotating when they are not or appear to not be rotating when they are. This is the same confusion some people have about the rotation of the moon - why we always see the same side of the moon, yet it is rotating. Just try making yourself a demonstration with a couple of pieces of fruit and you'll get it.
Or do you mean how did the earth get its tilt in the first place? No one knows for sure - perhaps a collision when it was young.
Impossible, or just very improbable? Since a circle is an elipse, I would have thought that it was no more unlikely than any other degree of eccentricity.
You are mistakenly assuming that just because Earth's periapsis passage currently is nearly coincident with the winter solstice that that is the way things have always been. That is not the case. The Earth's anomalistic year (the time between successive periapsis passages) is 25.1 minutes longer than the Earth's tropical year (he time between successive vernal equinoxes). That the Earth's periapsis passage currently is nearly coincident with the winter solstice is a coincidence. A couple of thousand years from the Earth's periapsis passage will happen to occur in February. A few thousand more, it will occur in April.
A circular orbit is a mathematically impossible in the sense that the probability of picking some exact number from a continuous random distribution with a continuous CDF is identically zero. To have a circular orbit the velocity vector must be exactly orthogonal to the position vector (an event of measure zero) and the magnitude of the velocity vector must have some exact value (another event of measure zero).
A circular orbit is mathematically impossible in a system with a rotating star because the star's gravitational potential is no longer just the inverse of the distance. The potential will depend on direction as well as distance because the rotating star will have non-zero higher order mass moments.
A circular orbit is mathematically impossible in a solar system with a more than one planet because those other planets will perturb each others' orbits.
An eccentricity can get close to 0, but when the decimal places examined approach infinity, the probability approaches 0.
But consider an orbit whose eccentricity oscillates from 0.01 to 0.02. During this oscillation, it must pass through every point in between, no matter how many decimal places we examine, hence every value in between is certain to happen, at least momentarialy. Since there is no such thing as a negative eccentricity, an oscillating eccentricity will never pass through 0.
Does the tilt have a period or a certain date it will be maximun or minimum?
Again: tilt is constant. Ie, no maximum or minimum.
No, it's not. The Earth's orientation changes with time; the changes are collectively called precession and nutation. The largest in magnitude and longest in period was first discovered by Hipparchus. Because the Moon's orbit about the Earth and the Earth's orbit around the Sun are inclined with respect to the Earth's equatorial plane and because the Earth is an oblate spheroid, the Moon and Sun exert torques on the Earth. The principal effect is the lunisolar precession, also called the precession of the equinoxes. The Earth's orientation approximately traces out a cone with a half angle of about 23.4 degrees (the obliquity of the ecliptic) and a period of 25,770 years.
Lesser processes are collectively called nutation. The nutation terms have a significantly shorter period and a significantly smaller magnitude than does the luni-solar precession. The largest magnitude nutation term has a period of 18.6 years, corresponding to the 18.6 year luni-solar gravity cycle. The Chandler wobble is a torque-free precession of the Earth. Except for this Chandler wobble, the luni-solar precession and various nutation terms are caused by external torques on the Earth. Some of the nomenclature exists because the processes were discovered and named before the discovers had a good understanding of why the processes exist.
There are also variations that have periods that exceed the 25,770 year luni-solar precession. The obliquity of the ecliptic itself is not a constant. It varies between 22.1 degrees and 24.5 degrees with a period of about 41,000 years.
I was still talking in the short term - month to month timeframe (from my previous post). It appears to me that the OP is asking about month-to-month changes. Over that timeframes it is effectively constant.
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