Also, this was my third or fourth attempt. The first ones were tumbling train wrecks. The derivation

Physicist : This question always bothered new england bed and breakfast inns me too.  The short answer is: it falls out of the math.  Specifically, the math of first year physics and second year calculus.  The fact that the Sun is in one focus is just one of those things.  It s nothing special.  Even less special is the other focus, which contains nothing at all.
I can t find a good intuitive new england bed and breakfast inns reason why orbits are elliptical.  In fact, I can t even find a mathematical derivation.  So, because it should be found somewhere, I ll leave the derivation new england bed and breakfast inns floating in the answer gravy.
Answer gravy : The force of gravity is usually written new england bed and breakfast inns as .  You can rewrite this using vector notation as , where the dot on top is a time derivative.  To keep the notation both standard and confusing, .
c is an integration constant , it can be any number.  Jumping new england bed and breakfast inns over to polar coordinates you can rewrite the usual velocity in terms of how fast you re moving toward or away from the Sun and how fast you re going around .
The choice of P and ε may seem arbitrary (and it is), but it has some historical relevance.  P is called the semi-latus recturn and it basically describes the size of the orbit.  ε is called the eccentricity , and it describes how lopsided the orbit is.  ε=0 means the orbit is a circle, 0 ε 1 means the orbit is elliptical, and 1≤ε means that the orbit is open (not actually orbiting).  For reference, the Earth s eccentricity is ε=0.01671123 and Halley s comet s is ε=0.967.
Standard and confusing is right. Someone must have been reading Jackson recently. My favorite David Jackson story: a friend of mine met him once at a conference new england bed and breakfast inns and asked him why he dedicated the book to his father. His reply? So everyone would know I had one.
Just a curious question. Is this one of those things that you explicated that is known or is it something that you can just figure out by looking at it HAVING NEVER SEEN IT. I m just curious. new england bed and breakfast inns I m a math and physics double major and I can follow the math (I m only in my first year . ) but I always feel stupid when someone new england bed and breakfast inns pulls out a huge derivation like this (don t get me wrong, I love them) but I m always like, is this something they thought of or is this like, written down? Should I be able to do this? I m not sure what I m trying to say I guess lol.
My thinking was: write down F=MA and go from there. Knowing that the end result is an ellipse means that the solution can be in a form that s independent of time (it s just a shape). I chose to used R and theta instead of x and y because the conservation of momentum has a simpler form, and it gives you a chance to get rid of the time dependence.
Also, this was my third or fourth attempt. The first ones were tumbling train wrecks. The derivations you see are almost never the first attempt, and generally it s streamlined so much that you can barely see what the author was thinking.
In your first year it would be unusual to be able to do this. Having experience seeing tricks other people use is in derivations is very important. When a professor is deriving stuff it helps (both of you) to ask why did you do that? or what is the goal in this step? . Harass them during office hours as often as you can, and keep a dialog open while they work through problems. It takes the mystery away from the whole process.
However, simple mechanics tells us that no free body can orbit around another moving body in closed geometrical path. [Central body of a planetary system is also a moving body]. Therefore, elliptical orbits are imaginary constructs, which may show relative positions new england bed and breakfast inns of concerned new england bed and breakfast inns bodies in space. They do not depict paths of these bodies in space. In reality there are neither elliptical orbits or foci for the central bodies to be in. Real orbital path of a planet is wavy about the path of its central body.
The calculation in this post assumes that the central mass is stationary, however in reality each body orbits the center of mass of the system, and both of those orbits are elliptical (with the center of mass in a focus).
For more than two bodies, and taking into account general relativity, you find that the orbits are only very, very, very nearly elliptical. The orbit of Mercury, due to general relativistic effects, shifts the farthest point in it s orbit by about 1 degree every 8.5 thousand new england bed and breakfast inns years (so it doesn t quite close).
Elliptical orbits are perhaps due to the fact that the universe is expanding! Thus, the expanding movement imparts an excentricity to orbits that would have been circular if the universe were stationary!
Good basic point in that it is the expansion of our universe that imparts movement on large heavenly bodies, thereby causing eccentric movements of lesser heavenly bodies new england bed and breakfast inns around these larger ones makes sense? The only difference I see would be if there is no expansion of the universe, new england bed and breakfast inns in which case there would be no significant movements (circular or eccentric) of bodies around their larger counterparts. In the sense of a retraction(collapse) of our universe I would think that everything new england bed and breakfast inns would then begin to operate in a reverse, perhaps even eccentric, order.
Even if you have two equal-mass new england bed and breakfast inns bodies they ll still follow elliptical orbits. The only change is that, instead of the Sun being in one focus it s the center of mass of both bodies. Technically, this is always the case, it s just that when the Sun is involved the center of mass and the center of the Sun are almost the same.
What I now gather is that the Sun is not at the center of an elliptical orbit, but rather.. it is a little off to one side; at a point called a focus of the ellipse (just as you stated earlier). And it is because of this offset, that causes a planet to move closer to(perihelion) and further away(aphelion) from the Sun every orbit. Thanks new england bed and breakfast inns for your time.
Veribion. What he said is that the Sun is not exactly at the focus, but orbiting it. The thing is that the focus is inside the Sun and very close to its center so its really jut waving a bit. This also means that we can approximate the Sun as being in the focus.
Q: Is it possible to fill a black hole? If you were to continuously throw galaxies worth of matter into a black hole, would it ever fill up? And what would theoretically happen if all the matter in the universe was thrown into a single black hole?
Q: If black holes are rips in the fabric of our universe, does it mean they lead to other universes? If so, then did time begin in that universe at the inception of the black hole? Could we be in a black hole?