Earth's elliptical orbit

Hi aranoff and welcome to these Forums!

The real question is: "Why is the Earth's orbit so nearly circular?"

A circle is the ideal limit, Limit e -> 0, of an ellipse. Keplerian motion, explained by Newton's laws, leads us to expect every freely orbiting body to orbit its parent body on an ellipse.

The fact that the Earth's orbit is so nearly circular, e = 0.017, speaks of the averaging of orbital elements of the millions of random collisions of the planetesimals that made up the Earth. The result was very nearly circular.

Proto-planets left with high ellipticity would also be ejected by close encounters with other planets. The eight planets plus dwarf planets left are the end result of such an accretion process.

Garth

 

Elliptical orbits are the norm and circular orbits are rare. All the gravitational nudges from the other objects in the solar system can disrupt a perfectly circular orbit. However, the Earth's orbit is fairly circular (http://www.seds.org/billa/tnp/help.html#eccentric [Broken] of only 0.02).

 

Since nobody has answered your question directly, I will. There is no tangential acceleration (at least, not in the simple two body problem of Newtonian mechanics). No tangential acceleration is needed.

Central force motion (systems in which force is a function of distance to the origin only) does not mean the body subject to the central force is constrained to move in a circle. When the force is proportional to 1/r^2, the body is constrained to move along a conic section (circle, ellipse, parabola, or hyperbola).

 

The acceleration is still directed against the position vector for an elliptical orbit (or a parabolic or hyperbolic orbit, for that matter). What makes you think there has to be a tangential acceleration?

I think you are going wrong by assuming that the velocity vector is normal to the position vector. For a non-circular orbit, the velocity vector almost always has a non-zero radial component. The velocity is normal to the position vector at apofocus and perfocus only.

 

It's fairly easy to show there is no tangential acceleration using cylindrical coordinates. The position of an orbital body with miniscule mass in cylindrical coordinates is

[tex]\vec r = r \hat r[/tex]

Differentiating with respect to time,

[tex]\dot{\vec r} = \dot r \hat r + r \frac d{dt}\hat r
= \dot r \hat r + r \dot{\theta} \hat{\theta}[/tex]

Differentiating once more to get the acceleration,

[tex]\ddot{\vec r} =
(\ddot r -r \dot{\theta}^2) \hat r +
(2\dot r \dot{\theta}+ r \ddot{\theta})\hat{\theta}[/tex]

The term [itex](2\dot r \dot{\theta}+ r \ddot{\theta})\hat{\theta}[/itex] is the tangential acceleration. The next few paragraphs show that this tangential acceleration is zero due to conservation of momentum.

The angular momentum with respect to the origin is

[tex]\vec h = \vec r \times \dot{\vec r} = r^2\dot {\theta} \hat z[/tex]

In any central force motion problem, the force on the body is, by definition, parallel to the position vector. There is no torque on the body since [itex]\vec \tau = \vec r \times \vec F = 0[/itex]. No external torques => constant angular momentum. Thus

[tex]\frac d{dt}(r^2\dot {\theta})
= r(2\dot r \dot {\theta} + r \ddot {\theta}) = 0[/tex]

and the tangential acceleration is zero.

 

Earth's eccentricity varies from nearly circular to about 0.06 in cycles of about 400,000 years. Jupiter is the main cause of this. So in addition to the large acceleration vector that is always directed radially towards the sun, perhaps Jupiter supplies the tangental component you're looking for.

 

This is incorrect.

I don't know what level of math or physics background you have. Solving the two body problem is covered in part in freshman-level physics classes and in full in upper-level undergraduate classes.

All you can say about a planet's orbit is that it is a conic section. A circular orbit is a very special case. Given the masses of the objects and the separation between them, there is only one velocity that results in circular motion. Any other velocity results in some other conic section.

 

Which is what I said.

And I concur that the presence of gas, dust and minor bodies adds to the process.

Garth

 

Aranoff's difficulties are much more basic than describing how addtional bodies perturb the Earth's orbit. He doesn't understand how the two body problem can result in non-circular motion. All: Please keep the perturbations due to multiple bodies or relativity out of this.

Aranoff: That the two-body problem can result in non-circular motion was known even to Kepler, in an ad-hoc manner. Newton developed the mathematics that describes how the two-body problem results in a conic section.

 

It's all in the initial conditions.

Unless the orbital velocity is initially exactly that of the circular velocity and initially exactly normal to the radial direction for both bodies then the two bodies will be on elliptical or hyperbolic orbits.

It's basic orbital dynamics, it is the exactly circular orbit that is impossible to achieve.

Garth

 

Aranoff, could you elaborate on why you think a non-circular orbit is impossible? (It is not impossible; as many have said, it is basic orbital mechanics. In fact, it is the circular orbit that is impossible to achieve.)

It would also help if you could let us know what level of mathematics and physics education you have received.