# Why wouldn't the Earth gravitate towards the Sun?

1. May 31, 2013

### Kyoma

Hello.

Why wouldn't the Earth gravitate towards the Sun? I have done some online research on this question and it appears that the reason is due to the Earth's rotation.

I am not convinced and see no reason why that should be the reason. According to the Rutherford's model of the atom, an electron (which has a spin) would spiral towards the nucleus and it is only when Bohr came along with the Bohr's model that the problem was solved: that electrons could only orbit in certain orbits.

Hence, it seems that this could be applied to the Earth-Sun model.

Much help would be appreciated ^^.

P.S. I have another question: why does the Earth rotate? Is there a reason for it or does it just happen naturally?

2. May 31, 2013

### jtac

The speed of the orbit of the Earth around the sun is what dictates it's distance to it. If the Earth would slow down it's orbit, it would fall closer in to the sun and vice versa, if it would speed up it would end up further away from the sun.

Rotation of the earth is a tougher question, I'm not sure where the rotational energy comes from originally, I guess it would be from the infall of matter as the earth formed and then stabilized by the orbit around the sun and the molten core, afaik all planets rotate, only moons that are tidally locked and asteroids and other smaller objects tumble or stay relatively stationary.

3. May 31, 2013

### D H

Staff Emeritus
The Earth does gravitate toward the Sun. If the Earth wasn't gravitating toward the Sun it wouldn't be orbiting.

The Earth's orbit is nearly circular and the speed is nearly constant, so let's treat it as such. That's uniform circular motion. Uniform circular motion is a kinematic description of the motion. Some attractive central force is needed to create that motion. The centripetal acceleration is given by $a=r \omega^2$. For an object undergoing uniform circular motion at a distance of one astronomical unit and an angular velocity of one revolution per sidereal year, this centripetal acceleration is 0.059301 m/s2.

That's a purely kinematic description of the Earth's orbit. It's accelerating sunward at a (nearly) constant 0.059301 m/s2. Let's see if gravity is what provides that centripetal acceleration. Per Newton's law of gravitation, the gravitational acceleration of the Earth toward the Sun is $GM_{\odot}/r^2$, where $GM_{\odot}$ is 132,712,440,018 km3. Once again using a radial distance of one AU, Newton's law of gravitation yields an acceleration of 0.0593008 m/s2. To five decimal places, this is the same number as the value needed to produce the observed centripetal acceleration. It's not an exact match because the Earth's orbit is not quite circular.

(Aside: It's much better to use μ, notionally the product G*M, as opposed to G*M. What's the difference? The gravitational parameter μ for some object can be observed by looking at how things orbit that object. Science knows the solar gravitational parameter to more than eleven decimal places. In comparison, the gravitational constant G and the solar mass M are known only to four each.)

You might be confusing rotation and revolution. The Earth revolves about the Sun, one revolution per sidereal year. The Earth rotates about its own axis, one revolution per sidereal day.

It just happened long ago, when the Earth was born. Angular momentum is a conserved quantity. This means that an external torque is needed to change angular momentum. Just as no external force means no change in velocity (Newton's first law), no external torque means no change in angular momentum.

One final note: There are external torques on the Earth by the Moon and the Sun. The tides are very slowly changing the length of a day. A "day" (one revolution about the axis) was only 4 to 6 hours long 4.5 billion years ago.

4. May 31, 2013

### Kyoma

Thank you for your replies.

But actually, I would like to ask why wouldn't the Earth move closer to the Sun? Since there is gravitational pull by the Sun on Earth, wouldn't Earth move closer to the Sun and be destroyed by the latter?? Is it due to the Conservation of Angular Momentum that is keeping the Earth from moving closer to the Sun?

I also realise that the Earth undergoes two kinds of angular momentum: one is revolution around the Sun and the other is rotation about its axis. But is this true?

Thank you. ^^

5. May 31, 2013

### bapowell

The Earth is always falling towards the Sun, but its angular momentum is such that there's always enough tangential velocity to keep Earth from falling into the Sun. The result is a periodic orbit.

6. May 31, 2013

### Ogr8bearded1

D H and bapowell both gave good answers. It is the Conservation of Angular Momentum that is keeping us from falling into the Sun as you asked. There are two kinds of angular momentum as you asked. Our rotation on our axis is slowing due to a transference of momentum to the Moon. Since the Moon is getting more momentum this causes the Moon to move further from the Earth and our rotation decrease. This does not cause the Earth to move closer to the Sun because when you consider the angular momentum of the Earth to the Sun you are really looking at the combined angular momentum of the Earth-Moon system. What is lost by the Earth is gained by the Moon so the Earth-Moon angular momentum is conserved in relation to the Sun.

Your other question about why the Earth rotates leaves me at a loss. I see only jtac tried to answer this, and while that sounds logical, it still leaves open the question of why does Venus rotate opposite of most of the other planets and why does Uranus have its North/South axis near where all other planets have their equator. What caused their material to infall differently from the others? What these two planets do show is that the angular momentum of spinning on the axis is indeed different from the angular momentum of the orbit.

7. May 31, 2013

### Mordred

The Earth rotates/spins for the same reason conservation of angular momentum. During Earths formation, the accumulated dust imparted inertia. Venus unusual rotation is due to being struck. As far as I understand

8. Jun 2, 2013

### D H

Staff Emeritus
Conservation of angular momentum and dust accumulation explains the rotation of Jupiter quite nicely. Jupiter is the 600 pound gorilla in our solar system. Saturn and Neptune? Maybe. Uranus? No. It's rotational axis is nearly orthogonal to it's orbital axis.

The standard hand wave explanation for the 97.86° obliquity of Uranus is that something big must have smacked it near the end of the formation of the solar system. In fact, two such giant impacts are needed. An alternate explanation requires no collisions: G. Boué and J Laskar (2010), A collisionless scenario for Uranus tilting, ApJ 712 L44. The obliquity of Uranus remains an open issue.

Conservation of angular momentum and dust accumulation does not explain the rotation of the inner planets. For one thing, dust accumulation is an overly simplistic model of how the inner planets formed. For another, planetary rotational angular momentum is not a conserved quantity for any of inner planets.

The gravity gradient (tidal) torque by the Sun on Mercury is huge. Mercury quickly transitioned to an equilibrium state (a stable spin-orbit resonance such as the current 3:2 spin-orbit resonance) shortly after it formed. Whatever rotational angular momentum Mercury had when it first formed is long lost. Mercury's rotational angular momentum has not been conserved.

In the case of Venus, a synchronous 1:1 spin-orbit resonance is unstable thanks to Venus's very thick atmosphere. Venus is in, or very close to, a stable rotational state. The planet is rotating retrograde but the upper atmosphere is rotating prograde. (Prograde with respect to the orbit, that is. Venus's upper atmosphere rotates retrograde with respect to the body.) This counter-rotating body, counter-counter-rotating atmosphere is one of the four four possible end states for Venus's rotation. Venus is in a stable rotational state. In a sense, it is tidally locked to the Sun.

For a wide variation of initial conditions, there are but four possible end states for Venus's rotation, two with the planet rotating prograde and two with it rotating retrograde. (A. C. M. Correia and J. Laskar (2001), The four final rotation states of Venus, Nature, 411 (6839), 767–770.) There were two pathways by Venus could have arrived in its current rotational state. One involves a gradual slowdown and then reversal of the rotation rate, the other involves a chaotic transition where the rotation axis essentially flips. Which pathway did Venus follow? There's no way to tell. The two retrograde states are indistinguishable. As is the case with Mercury, whatever rotational angular momentum Venus did have when it first formed is long lost. Venus's rotational angular momentum has not been conserved.

What about Mars? "The numerical integration shows that the obliquity of Mars undergoes large chaotic variations. These variations occur as the system evolves in the chaotic zone associated with a secular spin-orbit resonance." (J. Touma and J. Wisdom (1993), The chaotic obliquity of Mars, Science, 259 (5099), 1294-1297.) Once again, invoking conservation of angular momentum to explain the rotation of Mars just doesn't work because Mar's rotational angular momentum is not a conserved quantity.

Finally, what about our own planet? Our own planet is the one planet where the giant impact hypothesis is the consensus view. (The giant impact hypothesis is not needed for Venus; the consensus view is that Venus's rotation is best explained by collisionless models. The giant impact model may not be needed for Uranus; see the start of my post.) Note very well: The Earth's rotational angular momentum is not a conserved quantity in the giant impact hypothesis. If this model is correct, more than 75% of the Earth's post-collision angular momentum has been transferred to the orbit of the Moon.

Bottom line: Except for Jupiter, and maybe Saturn and Neptune, planetary rotational angular momentum is not a conserved quantity. There are multiple mechanisms by which rotational angular momentum of a planet can be transferred to other bodies.

Last edited: Jun 2, 2013
9. Jun 2, 2013

### Chalnoth

I think the bigger point to be made is just that some degree of rotation is the normal state of affairs: it is almost impossible to collapse a large dust cloud and end up with a body that isn't rotating. Other effects can (and do) change the rotation of bodies later. But rotation is normal.

The questions about rotation get much more interesting when we ask why a body is rotating in a particular way (e.g. Uranus), or why it is rotating so slowly (e.g. Venus), or why it rotates so that it always shows the same face to the body it is orbiting (Mercury, our Moon, some other moons).

10. Jun 2, 2013

### Mordred

I didn't go into a lot of detail of other later factors. As Chalnoth pointed out rotation is normal, as the dust condensed the angular momentum is conserved. Later factors will alter the resultant rotation. Such factors include effects from other rotating bodies such as moons. (tidal locking) Impacts from other
bodies and proximity to the Sun again tidal locking.
These are later influences although proximity to the sun does have an initial effect mainly in the momentum of the dust swirls at the time of formation.

11. Jun 4, 2013

### D H

Staff Emeritus
It's fine not to go into details. What I'm objecting to is that "dust collapse" is not how planets form. It is how stars and the protoplanetary disk form, but not planets. Planet formation is a much more complex and not completely understood problem.

In the context of a student asking why the Earth rotates, don't go back 4.6 billion years. The student most likely has the misconception that some force is needed to make the Earth rotate. The first and foremost thing to do is to dispel that notion. The simple answer is that the Earth is rotating today because it was rotating yesterday. Over the short haul, a planet's rotational angular momentum is conserved. Go beyond yesterday and things starts to get complicated. 300 years ago, the mean solar day was 86400 seconds long. Today the corresponding figure is 86400.002 seconds. The Earth is slowing down because external torques are acting on the Earth. Go even further back and the change is more pronounced. For example, a day was only 22 hours long 600 million years ago.

Going all the way back to the Earth's origin: I'd say the best thing is not to do that. At some point, the best answer to a student who unknowingly asks a question with an inherently complicated answer is to say so. Don't give an answer that is ultimately incorrect. When a student asks where those torques originate, the answer is that almost all of it comes from the ocean tides. When the student presses, the correct answer is "Well now you are asking a very complicated question." Invoking the tidal bulge is wrong because there is no such thing, at least not in the ocean tides. The oceanic tidal bulge does not and can not exist. It's a fundamentally flawed explanation.

As is dust collapse.

12. Jun 4, 2013

### Mordred

Last edited by a moderator: May 6, 2017
13. Jun 4, 2013

### Chronos

The main stream consensus supports the idea that planets are conferred angular momentum by their birthing protoplanetary disks. It is, however, a process fraught with uncertainties that are usually not addressed in lay articles. A review of theoretical planetology [e.g., Building Terrestrial Planets, http://arxiv.org/abs/1208.4694] illustrates the complexities involved and why it is difficult to ascertain when, how, or how much angular momentum is transferred to planets from a protoplanetary disk. We can safely conclude the particles that comprised the initial protoplanetary disk were travelling in the same direction, but, little more. Logic driven conclusions are seductive, but, not always reliable.

Last edited by a moderator: Jun 4, 2013
14. Jun 4, 2013

### Chalnoth

This discussion is becoming overwrought with pedantry.

Again, two main points:
1. When you have a large, diffuse cloud of gas and dust collapsing inward, the normal state is rotation.
2. The precise way in which the gas and dust that formed the Solar system collapsed was exceptionally complicated and the details are not fully-understood. Later events can also dramatically change the rotation of these bodies in complicated ways.

The first point shows that, "Why are the planets rotating?" can be answered trivially: rotation is the default. We should expect rotation. That is to say, we should not be surprised when we see a planet rotating.

The second point shows that, "Why are the planets and moons rotating in the precise way in which they are rotating?" is a horribly complicated question to answer, and that answer is not yet fully known.

Not having the full answer to the second question doesn't mean we don't have an answer to the first (we do).

Last edited by a moderator: May 6, 2017
15. Jun 5, 2013

### Chronos

Your smug self assurance is annoying, chalnoth.

16. Jun 5, 2013

### Ibix

A point that I haven't seen addressed is your analogy with the Bohr atom.

Bohr's orbits were problematic because we know that an electron going round in a circle will emit electromagnetic radiation. The energy radiated has to come from somewhere, and it comes from the electron slowing down. That means that the electron spirals in - or it would if it weren't for quantum effects.

Planets aren't charged, and their orbital periods are much longer than atomic orbits, so they don't lose energy to electromagnetic radiation in the same way. They do lose energy to gravitational radiation, but the effect is minuscule. We have managed to see it happening once under really extreme circumstances - colliding neutron stars. So yes, we spiral in. But very very slowly.

17. Jun 6, 2013

### Chalnoth

I'd be willing to bet that for planets in our solar system, other effects are larger than gravitational radiation. Other things that can impact a planet's orbit:

1. Friction through the small amount of gas and dust in our galaxy.
2. Decreases in the Sun's mass over time (the Sun loses mass through the radiation it outputs and through the solar wind....this mass loss will increase in the future).
3. Tidal forces which tend to make it so that bodies which rotate faster than their relative orbits move further away from one another (e.g. the Earth and the Moon will continue to get further away until the Earth always shows the same face to the Moon).

All of these effects are minuscule effects, and I think the only one that will become noticeable for the Earth's orbit is the mass loss of the Sun, but that probably won't be noticeable for billions of years yet (possibly not until our Sun goes red giant, or is close to doing so).

There may also be other effects that I'm not aware of, but I'd be willing to bet that gravitational radiation is probably less than any of these for the Earth and most other planets. This certainly isn't true for all gravitational systems, of course. Binary neutron stars are a prime example.

18. Jun 6, 2013

### D H

Staff Emeritus
Correct on both accounts. Gravitational radiation is an incredibly tiny and unobservable effect for the Earth, as it is (AFAIK) for every other body in the solar system. The Earth apparently is slowly spiraling out from the Sun, not in, at an apparently anomalistically high rate of 15±4 meters/century. Note well: I said "apparently" twice.

19. Jun 7, 2013

### Ibix

I stand corrected.

The reason that we would (naively) expect electrons to spiral in is EM radiation emission. Although there is an analogous mechanism for orbitting bodies to emit gravitational radiation, the effect in the case of the Earth is too small to measure (even with the remarkable precision D H cites), not just minuscule.

20. Jun 7, 2013

### Chalnoth

Incidentally, some discussion of the gravity wave amplitude from the Earth-Sun system:
http://en.wikipedia.org/wiki/Gravitational_wave#Wave_amplitudes_from_the_Earth.E2.80.93Sun_system

(As I link this, the equations don't show up on my screen...but hopefully that's either a temporary problem or one limited to my computer)