Elliptical orbits

[Stargazer_10]

Just wondering why planets orbit the sun elliptically? Why is it not circular?
Thanks for the help in addition.

[mithrandir (Andrew)]

Originally Posted by Stargazer_10

Just wondering why planets orbit the sun elliptically? Why is it not circular?
Thanks for the help in addition.

A circle is the special case of an ellipse where it has eccentricity zero. The general formula for an ellipse is

(x/a)^2 + (y/b)^2 = 1 with a>=b

The eccentricity is e = sqrt(1-(b/a)^2) and will be less than 1

If a and b are equal e will be zero. As a and b are equal we can rename them both in the formula as "r". Multiply both sides by r^2 and you get the usual formula for a circle

x^2 + y^2 = r^2

Even if a planet at some stage had a circular orbit, perturbations from the gravity of all the other objects would very quickly change the orbit to make it elliptical.

[LAW (Murphy)]

This isn't exactly comprehensive, but it's not a bad start.

http://www.youtube.com/watch?v=uhS8K4gFu4s

Supplementarily...

http://www.youtube.com/watch?v=danYFxGnFxQ

Further more...

https://encrypted-tbn0.gstatic.com/i...7Lge3lU4OPKHoA

[Wavytone]

You should study Newton's 'Principia', all is revealed. I have a copy, it is an amazing read and a magnificent glimpse into a beautiful mind.

You will need to be good at maths, specifically calculus at HSC (year 12) level.

Newton starts with the basic laws of motion then the postulate for gravity. From this the proof of elliptical orbits follows in straight word fashion. There are some interesting examples along the way, regarding the moon and tides as well.

It is available online as (obviously) its out of copyright.

It also shows how complex calculations were done in newtons day, and shows how the methods taught to kids these days have been dumbed-down and are quite primitive, thanks to calculators. Newtons methods are described in a book called 'The Trachtenberg System of Speed Mathematics'. Trachtenberg merely appropriate the methods Newton used and published under his own name. It's also the method that so-called 'geniuses' use to perform calculations on stage and TV.

[sjastro]

To answer the question one needs to ask what forces act on an orbiting body.
In the Newtonian model these forces are a centripetal force acting along the radius of the orbit and a tangential force.
The orbit is essentially a balancing act between the forces.

The existence of the tangential force depends on the angular velocity of the orbiting body not being constant. (The angular velocity is the number of revolutions per unit time.)

It can be shown using high school maths for circular motion, there is no tangential force acting on an object if the angular velocity does not change. So in the absence of external forces acting tangentially on the object such as the thrusters of an artificial satellite or the gravitational pull of an external body, a circular orbit is unstable.

For an elliptical orbit tangential forces do exist for the simple reason the radius or distance between the orbiting and central body varies along the orbit. The angular velocity changes according to the distance as defined by Kepler's laws of planetary motion.

Note that the Newtonian model is only an approximation. A much more detailed explanation can be given using General Relativity.
General Relativity helps to explain why Mercury's orbital plane also rotates around the Sun, a feature unexplained by Newtonian gravity.

Regards

Steven

[bloodhound31]

Originally Posted by LAW

This isn't exactly comprehensive, but it's not a bad start.

http://www.youtube.com/watch?v=uhS8K4gFu4s

Considering the OP has a post count of 10, I'd say he's relatively new so this video is probably the best one of the lot. Short, sweet and to the point without all the techno-babble and formulae.

Thanks for that.

Baz.

[sjastro]

Originally Posted by bloodhound31

Considering the OP has a post count of 10, I'd say he's relatively new so this video is probably the best one of the lot. Short, sweet and to the point without all the techno-babble and formulae.

Thanks for that.

Baz.

Except it doesn't answer his questions.

[LAW (Murphy)]

Yeah, I don't think NASA will be using my post to map spiral arm evolution but I was hoping to just give a basic idea. Given the simple nature of the question (and the complex nature of the answer) I picked cartoons over calculus to get the point across.

[bojan]

You do not need calculus to understand better why orbits are oblate (elliptical).

A loooong time ago I wrote a computer program (in BASIC, for ZX-81 computer.... or was it AppleII?) and simple graphics to demonstrate the idea (it was inspired by the book written by well known Russian astronomer Yakov Preljman (at least he is known to me..) where he presented the simple calculation and planet orbit graphs.. [btw, this method is (badly and quite insufficiently) hinted in the video clips, peppered quite un-necessary by n-dimensional space later in comments, to add to the confusion].

Pereljman used a piece of paper to draw the orbit, segment by segment, calculated by a very simple algorithm, as below:

1) The Sun is stationary in the centre, and it's gravity causes the acceleration of a planet according to its distance from the Sun (and Sun's mass, of course) towards it.
2) Planet itself is positioned at a certain distance from the Sun, and has a certain arbitrary velocity of it's own, in arbitrary direction
3) After a certain period of time, the planet moved to the new position, because of it's own velocity and acceleration caused by the Sun.
4) This new resulting position and velocity is used for the next round of calculations (in programmers words, GOTO 1) )

If the time segments are chosen small enough (which is what calculus actually does, with time steps getting closer and closer to zero), the final graph on the screen will be very close to ellipse (or parabola) with the Sun in one of the focii and it will be stable for many orbits.. and of course if the initial chosen velocity of the planet is not too large or too small.

The algorithm demonstrates that inverse square law of gravitation (used in simulation to calculate the position of planet after time interval) has the property of causing the closed elliptical orbits (including circle and parabolic/hyperbolic orbits as special cases).

Eventually the orbit will change with time but in this simulation the cause is finite time interval (the code is not calculus!) and rounding errors (BASIC was using 7 or 8 digits, not enough precision for calculations like this one..).

I will try to find that code... or I will write a new one when I have time, it is really very simple job to do, just a couple of lines and a program loop, with plotting a dot on the screen.

Edit:
Link to the abovementioned book is here: http://archive.org/details/AstronomyForEntertainment
Pages 170 onwards.

[barx1963 (Malcolm)]

The simplest explanation I have heard is that there are an infinite number of possible orbits that a planet of given mass and speed can adopt, all of which are ellipses. Only 1 of those orbits as actually circular ( the circle being a special case of a ellipse with both focii being in the same place).
So while a perfectly circular orbit is possible, it is highly unlikely.

In a simply 2 body system, it is more likely that the orbit will be highly elongated, again simply because there are far more possible orbits that are highly elongated than ones that are nearly circular. And this is what we see with double stars where elongated orbits are the norm.
In a system with multiple planets the interactions of the various bodies combine to make the plantary orbits more circular over time, hence the planets of our system have only small elongations. For example the Earth eccentricity is very small. The planet with the largest eccentricity is Mercury, which fits as it would be the least influenced by the interactions of the other planets relative to the Sun.

Hope this helps in a non technical way to explain it as it was explained to me.

Cheers

Malcolm

[bigjoe (JOSEPH)]

Originally Posted by Stargazer_10

Just wondering why planets orbit the sun elliptically? Why is it not circular?
Thanks for the help in addition.

This might help. http://curious.astro.cornell.edu/que...php?number=467

CHEERS bigjoe

[sjastro]

Bodies in circular orbits are rare because they are sensitive to the gravitational effects of other objects.

Rather than going into the mathematics and being accused of engaging in "technobabble" here is a simple analogy.

An old party trick is to balance a chair on one leg. While it is difficult it is possible. From a physics viewpoint, the balanced chair is said to be in a state of "unstable equilibrium". The slightest nudge and the balance is destroyed and the chair either topples over or ends up resting on four legs.
If you apply the same nudge to a chair resting on four legs, it might rock about but ends up in its original configuration.

An object in a circular orbit is like the chair balanced on one leg. If you apply a perturbation the orbit is destroyed (chair topples over) or it ends up in closed elliptical orbit (chair rests on four legs).
The elliptical orbit on the other hand is far more stable to perturbations.

Incidentally the chap in the video who is confused about relating the dimensionality of space to the strength of gravity at least got the marble analogy in the bowl right, it applies to both orbits and chairs.

Regards

Steven

[tel.lekatsas (Tel)]

The guy in the video is not confused, there is a relationship between gravity and the number of spatial dimensions.

http://physics.stackexchange.com/que...-stable-orbits

[sjastro]

Originally Posted by tel.lekatsas

The guy in the video is not confused, there is a relationship between gravity and the number of spatial dimensions.

http://physics.stackexchange.com/que...-stable-orbits

If this idea is correct then physicists have been wasting their time since the early 20th century attempting to unify gravity with the other forces. A common feature with these theories (Kaluza-Klein and String Theories) is the extension of gravity into higher dimensions that allow unification to occur. Clearly stable orbits in higher dimensions occur in these theories.

Then there is the case of observation. The rotation curves of galaxies indicate that stars well away from the galactic centre do not exhibit familiar Keplerian orbits.
This indicates that the inverse square law doesn't apply either due to the presence of dark matter perturbing the stable orbits, or gravity doesn't need to be based on the inverse square law for stable orbits to exist.
In either case the observation contradicts the relationship between dimensionality and gravity.

Regards

Steven

[tel.lekatsas (Tel)]

Originally Posted by sjastro

If this idea is correct then physicists have been wasting their time since the early 20th century attempting to unify gravity with the other forces. A common feature with these theories (Kaluza-Klein and String Theories) is the extension of gravity into higher dimensions that allow unification to occur. Clearly stable orbits in higher dimensions occur in these theories.

Steven,

It depends on whether these extra dimensions are compact or not. If the extra dimensions are not compact then there are no stable orbits. On the other hand if the extra dimensions are compact there is no effect on the force law. KK and string theory assume compact extra dimensions.

Originally Posted by sjastro

Then there is the case of observation. The rotation curves of galaxies indicate that stars well away from the galactic centre do not exhibit familiar Keplerian orbits.
This indicates that the inverse square law doesn't apply either due to the presence of dark matter perturbing the stable orbits, or gravity doesn't need to be based on the inverse square law for stable orbits to exist.
In either case the observation contradicts the relationship between dimensionality and gravity.

Regards

Steven

The rotation curves of galaxies don't invalidate the inverse square law. The inverse square law can result in non-Keplerian orbits depending on the form of the gravitational potential and thus ultimately on the mass distribution within the galaxy. An exponential disk potential amongst many others does not give a Keplerian radial profile.

Kepler orbits arise from spherically symmetric static potentials.

Cheers,

Tel

[sjastro]

Originally Posted by tel.lekatsas

Steven,

It depends on whether these extra dimensions are compact or not. If the extra dimensions are not compact then there are no stable orbits. On the other hand if the extra dimensions are compact there is no effect on the force law. KK and string theory assume compact extra dimensions.

Tel,

You seem to be trying to reconcile Newtonian gravity into String Theory.
It doesn't work. Newtonian gravity is scale independent, the inverse square law applies to both small and large scales. Not so with gravity in String Theory.
In fact String Theory predicts the inverse square law is violated at small scales due to the existence of extra (compact) dimensions.

The rotation curves of galaxies don't invalidate the inverse square law. The inverse square law can result in non-Keplerian orbits depending on the form of the gravitational potential and thus ultimately on the mass distribution within the galaxy.......

A typical problem posed to maths and physics undergraduates is to prove that Keplarian elliptical orbits can only arise when the central force is based on the inverse square law.
As a corollary to this proof one can show that non Keplarian orbits can only arise if the central force is not a "pure" inverse square law.
Keplarian orbits arise under conditions such as spherical symmetry of the field and the absence of external forces.
If these conditions are not met the orbits are non Keplarian and can be derived through perturbation theory. In this case the perturbation results in additional central force terms that are smaller than the unperturbed inverse square term.
The central force is however no longer based on a simple inverse square law.

Regards

Steven

[bojan]

Originally Posted by tel.lekatsas

The inverse square law can result in non-Keplerian orbits depending on the form of the gravitational potential and thus ultimately on the mass distribution within the galaxy. An exponential disk potential amongst many others does not give a Keplerian radial profile.
Kepler orbits arise from spherically symmetric static potentials...

Originally Posted by sjastro

As a corollary to this proof one can show that non Keplarian orbits can only arise if the central force is not a "pure" inverse square law.
Keplarian orbits arise under conditions such as spherical symmetry of the field and the absence of external forces.
If these conditions are not met the orbits are non Keplarian and can be derived through perturbation theory. In this case the perturbation results in additional central force terms that are smaller than the unperturbed inverse square term.
The central force is however no longer based on a simple inverse square law.

Regards

Steven

I think you guys are talking about the same issue and have the same arguments, only using slightly different words..