# Earth's elliptical orbit

Why is the earth's orbit elliptical? That is, what force acts on the earth perpendicular to the force of the sun, that causes the tangential accelerations characteristic of an ellipse?

Garth
Gold Member
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

Phobos
Staff Emeritus
Gold Member
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).

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D H
Staff Emeritus
Why is the earth's orbit elliptical? That is, what force acts on the earth perpendicular to the force of the sun, that causes the tangential accelerations characteristic of an ellipse?

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).

earth's orbit

There is indeed a tangential acceleration. Acceleration is change in velocity. For a circular orbit, the change is perpendicular to the velocity, and parallel to the radius. For an elliptical orbit, the velocity changes in magnitude, and so the acceleration has a component perpendicular to the radius.

Where am I going wrong?

Thanks.

D H
Staff Emeritus
There is indeed a tangential acceleration. Acceleration is change in velocity. For a circular orbit, the change is perpendicular to the velocity, and parallel to the radius. For an elliptical orbit, the velocity changes in magnitude, and so the acceleration has a component perpendicular to the radius.

Where am I going wrong?

Thanks.

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.

D H
Staff Emeritus
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

$$\vec r = r \hat r$$

Differentiating with respect to time,

$$\dot{\vec r} = \dot r \hat r + r \frac d{dt}\hat r = \dot r \hat r + r \dot{\theta} \hat{\theta}$$

Differentiating once more to get the acceleration,

$$\ddot{\vec r} = (\ddot r -r \dot{\theta}^2) \hat r + (2\dot r \dot{\theta}+ r \ddot{\theta})\hat{\theta}$$

The term $(2\dot r \dot{\theta}+ r \ddot{\theta})\hat{\theta}$ 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

$$\vec h = \vec r \times \dot{\vec r} = r^2\dot {\theta} \hat z$$

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 $\vec \tau = \vec r \times \vec F = 0$. No external torques => constant angular momentum. Thus

$$\frac d{dt}(r^2\dot {\theta}) = r(2\dot r \dot {\theta} + r \ddot {\theta}) = 0$$

and the tangential acceleration is zero.

tony873004
Gold Member
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.

"Perhaps Jupiter..."

Has anyone done any calculations on the Jupiter effects?

earth's orbit

Can one explain why the orbit is an ellipse rather than a circle without invoking other planets?

russ_watters
Mentor
It is true that Jupiter has an effect, but you don't need another object in order for an orbit to be elliptical.

What you are missing, aranoff, is that the speed is changing, but the direction is not always tangential, so the acceleration is always directly toward the center of mass.

An elliptical orbit is the norm - a circular orbit is a special (read: impossibly specific) case of an elliptical orbit.

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Let me phrase my comments in another form.

Consider the two-body problem of the earth and sun, each replaced by point masses. Define let the coordinate system so that the orbit is in the x-y plane. At t=0, let the earth be at its closest distance, and moving in the y direction. Solve the equations of motion, using the initial conditions. The solution is a circular path.

The reality is that the orbit of the earth is an ellipse, with an eccentricity ε, which is a function of time. Since the fact of ε(t) is a fact that does not occur in the initial equations, there is no way to find a solution with ε ≠ 0.

I believe that the eccentricity is a result of the many-body problem, taking into account the large planets. There is no analytical solution to the Newtonian gravitational many-body problem. I do not have the tools to solve this problem. I wonder if any of you know about many-body calculations that result in the correct ε(t)?

russ_watters
Mentor
You've set up the problem in such a way as to demand a circular orbit, then asked why the orbit is circular. What if the moon was also moving in the X direction at t=0? What shape would you get then?

D H
Staff Emeritus
Let me phrase my comments in another form.

Consider the two-body problem of the earth and sun, each replaced by point masses. Define let the coordinate system so that the orbit is in the x-y plane. At t=0, let the earth be at its closest distance, and moving in the y direction. Solve the equations of motion, using the initial conditions. The solution is a circular path.

The reality is that the orbit of the earth is an ellipse, with an eccentricity ?, which is a function of time. Since the fact of ?(t) is a fact that does not occur in the initial equations, there is no way to find a solution with ? ? 0.

I believe that the eccentricity is a result of the many-body problem, taking into account the large planets. There is no analytical solution to the Newtonian gravitational many-body problem. I do not have the tools to solve this problem. I wonder if any of you know about many-body calculations that result in the correct ?(t)?

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.

As a researcher in the N-body problem I welcome your query. You are right to pose the problem. Let me try to make some comments.

Suppose we take Earth at present solar distance and add just Jupiter as perturbing body. This is a stable and well behaved system. Assuming we start with circular Earth, the subsequent behaviour would reveal oscillations. I have not done this recently but would guess eccentricity would reach a maximum around 0.05 or so. Adding Saturn would introduce some irregularity but giving essentially similar result.

The actual situation is complicated by the origin of the Solar System. It is generally accepted that the terrestrial planets formed by collisions of smaller bodies. The last few major collisions would be crucial for determining the Earth's starting eccentricity. And it is even more complicated since we believe the orbit may have become more circular due to the presence of gas and accumulation of minor bodies.

I hope the above is helpful. Please don't hesitate to come back to me.

Best regards, Sverre

Garth
Gold Member
Which is what I said.

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

Garth

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D H
Staff Emeritus
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.

No, I do not understand how a two body problem can give non-circular orbits, for then the result is more than the input.
It is not possible to get more out of a set of equations than the stuff that goes into the equations and initial conditions.

Garth
Gold Member
No, I do not understand how a two body problem can give non-circular orbits, for then the result is more than the input.
It is not possible to get more out of a set of equations than the stuff that goes into the equations and initial conditions.

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

D H
Staff Emeritus
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.

I think I got it. Consider my setup: A mass m is orbiting in the xy plane. At time t=0, the mass is at (x,0). It has a velocity v in the y direction. This is the given. Question: What will the orbit be?

For a certain velocity, the orbit will be a circle. Smaller velocities will not orbit. Larger velocities will be elliptical (or hyperbolic, for very large velocities).

Can one write the eccentricity as a function of velocity - velocity(circle), derived from the equation of motion?

My level: Ph.D. theoretical physics. I have been out of the field for a while, coming back to substitute in high school. My question arose from a student's question.

BTW, how do I write subscripts in these messages?

D H
Staff Emeritus
I think I got it. Consider my setup: A mass m is orbiting in the xy plane. At time t=0, the mass is at (x,0). It has a velocity v in the y direction. This is the given. Question: What will the orbit be?

Suppose the mass of the orbital body $m$ is miniscule compared to the mass of the central body $M$ (i.e., $m<<<M$). The circular orbit velocity is

$$\vec v_{\text{circ}} = \sqrt{\frac{GM}r}\hat y$$

What happens for other velocities?
• A parabolic orbit results if $\vec v = \surd2\,\vec v_{\text{circ}}$.
• A hyperbolic orbit results if the velocity is greater than the parabolic orbital velocity.
• An elliptical orbit results for all other velocities, including velocities lower than the circular velocity. If the velocity is larger than the circular velocity (but smaller than the parabolic velocity), the initial point is the perifocus of the orbit. If the velocity is smaller than the circular velocity, the initial point is the apofocus.

Can one write the eccentricity as a function of velocity - velocity(circle), derived from the equation of motion?

This is the specific energy of the orbit,

$$\frac{v^2}2 - \frac{GM}r = -1/2\left(\frac{GM}h\right)^2(1-e^2)$$

My level: Ph.D. theoretical physics.

Did you sleep through your junior-level classical mechanics class?

BTW, how do I write subscripts in these messages?

Write your equations in LaTex, as [noparse]$$LaTeX$$[/noparse], or use vBcode, [noparse] subscript [/noparse].

http://www.braeunig.us/space/orbmech.htm

The eccentricity e of an orbit is given by

e = Rp x Vp2 / GM - 1,

where Rp is the periapsis, and Vp is the velocity.

This is the answer to the question I posed: Given the periapsis and the velocity at this point, find the eccentricity.

I cannot find my old mechanics book. I had to search on line. But thanks to you for your comments, which helped me direct my search.

-Sanford

earth's orbit is elliptical because in December when earth is farthest from sun it is attracted towards galaxy for which Sun & entire solar system itsef has an orbit !

ideasrule
Homework Helper
First, this thread was started in April 2007. The Earth has followed its elliptical orbit about the Sun almost three times since then. Second, Earth is farthest from the Sun in July, not December. Third, the fact that the solar system is attracted to the galaxy is irrelevant because the acceleration of gravity is independent of mass, so all objects in the solar system would have the same acceleration. Fourth--and this point has been discussed to death already by more than one poster--Newton's second law and the inverse square law of gravitation lead DIRECTLY to the equation of an ellipse.

ok fine ! earth is farthest in july and not in december but again it has a co-relation with galactical gravitational force ! as earth aligns itself with the center of galaxy and sun twice in July and December the farthest and almost nearest and aligns with Solar system's locus/path in April and october ! dont you still see a co-relation in earth's elliptical orbit behaviour because of galaxy or it is merely a "co-incidence" !!!! you need to empty the cup ( filled with so many thumb rules) before putting the new principles. we're not kids , if a lesson is learnt it is universal every law/principles has some conditions but putting/converting into a predicate logic all the conditions need to be considered as well plus open up mind to at least consider if not adapt in entiriety . philosophy and science go hand in hand. Phd in science !! uh

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ideasrule
Homework Helper
ok fine ! earth is farthest in july and not in december but again it has a co-relation with galactical gravitational force ! as earth aligns itself with the center of galaxy and sun twice in July and December the farthest and almost nearest and aligns with Solar system's locus/path in April and october ! dont you still see a co-relation in earth's elliptical orbit behaviour because of galaxy or it is merely a "co-incidence" !!!! you need to empty the cup ( filled with so many thumb rules) before putting the new principles. we're not kids , if a lesson is learnt it is universal every law/principles has some conditions but putting/converting into a predicate logic all the conditions need to be considered as well plus open up mind to at least consider if not adapt in entiriety . philosophy and science go hand in hand. Phd in science !! uh

Perihelion is on January 3, while the Sun roughly aligns itself with the galactic center on December 18. That's a difference of half a month. If the difference was only a few hours, I'd find it interesting, but 17 days?

Besides, Earth's orbit changes; its perihelion precesses, revolving once every 110 000 years relative to the stars. The Sun is also orbiting the galactic center once every 240 million years. The fact that the Earth is close to perihelion when the Sun is roughly aligned with the galactic center means that we happen to be in the right portion of Earth's history for this to happen; it's a mundane coincidence, not something significant.

this 17 days gap is negligible! since this is a minute period in terms of "alignment" with galactic center as lights from different parts of galaxy travels on different intervals and by the time we measure the location of entire solar system which is again moving on a faster rate (red color means moving away from us , and blue means coming closer to us ) at the speed of light.

STARMAN :
"It's like in the highway where traffic is going only one way. The car in front of us is assumed not to be backing up into us as we see it approaching when it slows down. It is we who are accelerating toward. Those in that car would assume that we are closing the distance since they are obviously going in the same direction we are.

The same applies to those stars moving in orbit around the hub but which are ahead of us. If they are blue shifted then we are the ones closing the distance not them. Those outdistancing the sun would appear red shifted. The sun overtaking a star would also register as blue shifted.

Others are red shifted as the sun leaves them behind due to its greater orbital velocity. Others moving at almost our same velocity would appear almost stationary with only very slight red or blue spectral shifts.

A more dramatic example: We see areas of the universe receding from us at near light speed. If indeed someone is looking in our direction from over there they would see us moving away from them at near light speed. But this is attributed not to the objects themselves but to the addition of or increase of space taking place between us caused by universal expansion. So neither are really moving at that speed by away via proper motion alone but are being separated as space itself expands at ever increasing speed-a speed thought not to be limited by the speed of light.

The blue shifted Andromeda galaxy is also a good example. We say it's moving in our general direct about 300 kilometers per second (186 miles/sec.) If we were over there we would see the Milky Way Galaxy moving toward us at the exact rate. But then again not all is confusion since Andromeda's angle of rotation is known and because of the individual stars proper motions can be inferred via their spectral shifts. The same principle is applied to any other light-emitting object."

D H
Staff Emeritus
ok fine ! earth is farthest in july and not in december but again it has a co-relation with galactical gravitational force !
No, it doesn't. The Earth's orbit precesses due largely to the gravitational influence of the other planets. (General relatively also plays role here, but a much smaller role than the simple Newtonian gravitational attraction by the other planets.) With respect to the fixed stars, the Earth's apsidal precession is one revolution per 112,000 years. Compare this to the rate at which the solar system orbits the galaxy: one revolution per 225 million years.

The galaxy as a whole has essentially zero effect on the Earth's orbit. Some numbers: The Earth orbits the Sun at about 1AU and completes an orbit in one sidereal year. That corresponds to a mean acceleration of about 6×10-3 m/s2. The Sun (along with the Earth) orbits the galaxy at about 26,000 light years and completes an orbit in about 225 million years. That corresponds to a mean acceleration of about 2×10-10 m/s2. So we are already talking about a tiny effect. However, the effect of the galaxy on the Earth's orbit about the Sun is much, much smaller than this. The Earth also orbits the galaxy. It is the galactic tidal gravitational force rather than the galactic gravitational force that affects Earth's orbit about the Sun. The acceleration from this galactic tidal force is incredibly small, about 2×10-19 m/s2. This is, for all practical purposes, a non-effect.

philosophy and science go hand in hand.
No, they don't.

Phd in science !! uh
Are you claiming to have a "Phd in science"? If so, I call BS.

Integral
Staff Emeritus
Gold Member