How Does Dark Matter Affect Orbital Dynamics in the Universe?

[ubavontuba]

I would like this thread to explore the asociative effects of both dark energy and dark matter in the universe.

I have recently had the priveledge of learning a lot of good information from the likes of SpaceTiger and Garth and would like to relate some of what I've learned to this topic.

Initially, I would like to clarify some basic assumptions. I will start with some questions regarding orbital dynamics.

A body in orbit essentially embodies Newton's law that a body in motion will stay in motion unless acted upon by an external force, right?

It's orbit is determined by the gravity field in which it is orbiting, right? Does the gravity of the orbiting body effect the relative orbital relationship between it and the primary mass? That is, if Venus was replaced by another solar mass, would it still proceed around the sun in one venus year?

 

[SpaceTiger]

A body in orbit essentially embodies Newton's law that a body in motion will stay in motion unless acted upon by an external force, right?

To a certain approximation, yes, a body in orbit is described by both Newton's Laws of Motion and Newton's Law of Gravitation. This approximation is (probably) valid for the vast majority of orbits that are of astrophysical relevance. Only very near a black hole or a neutron star do relativistic effects unambiguously become important.

It's orbit is determined by the gravity field in which it is orbiting, right?
Does the gravity of the orbiting body effect the relative orbital relationship between it and the primary mass?

Yes. In this case, both bodies will be deep in a gravitational potential well and so both will be in motion. The two-body problem, however, can be neatly described as a one-body problem in a time-independent potential using the reduced mass. Thus, the results are qualitatively the same to a problem with one body of negligible mass in orbit around a much more massive one (like a planet around the sun).

That is, if Venus was replaced by another solar mass, would it still proceed around the sun in one venus year?

Actually, it would be eight and a half months. Kepler's Law is given by:

$$P^2=\frac{4\pi^2 a^3}{GM_{tot}}$$

where P is the period, a is the separation of the two bodies (for a circular orbit), and M_tot is the total mass of the system. Since the total mass is twice as large when Venus is replaced by a solar mass, the period is reduced by a factor of square root of two.

This is standard fare and can be found at the beginning of orbital dynamics texts. Also, it often gets chapters in classical mechanics books, so I would check those sources if you're interested in the derivations.

 

[Janus]

Actually, it would be eight and a half months.

Slight correction here. That should be about 5 and a quarter months, As the Venerian year is less than 7 and a half months long to start with. (Unless SpaceTiger was talking "Vererian Months" with each being 1/12th of a Venerian Year)

 

[SpaceTiger]

Slight correction here. That should be about 5 and a quarter months, As the Venerian year is less than 7 and a half months long to start with. (Unless SpaceTiger was talking "Vererian Months" with each being 1/12th of a Venerian Year)

You're right, just a bit of sloppiness on my part. It's about 8 and a half "Venus months", which equates to 5 and a quarter months on the calendar.

 

[ubavontuba]

Okay then, would it be correct to state that we now know that increasing the mass of the system increases the total gravitational potential and we thus have a corresponding increase in the orbital rate at a given distance?

How about in the case where we divide our super-sized venus into two equal masses placed exactly opposite to each other around the sun (but in the same orbit)? Of course this would mitigate the wobble in the system, but would we still have the same rotation rate for our two venuses about the sun?

 

[SpaceTiger]

Okay then, would it be correct to state that we now know that increasing the mass of the system increases the total gravitational potential and we thus have a corresponding increase in the orbital rate at a given distance?

With spherical symmetry, that should be the case, yes.

How about in the case where we divide our super-sized venus into two equal masses placed exactly opposite to each other around the sun (but in the same orbit)? Of course this would mitigate the wobble in the system, but would we still have the same rotation rate for our two venuses about the sun?

No, three-body systems are considerably more complicated than two-body systems. In fact, the setup you're suggesting can be chaotic, sometimes leading to ejection of one of the bodies. The reason we can even try to model things like galaxies is that the stars are well outside the tidal spheres of the other stars in the galaxy. In this limit, we treat the star as a test particle moving through a time-steady gravitational potential.

 

[ubavontuba]

No, three-body systems are considerably more complicated than two-body systems. In fact, the setup you're suggesting can be chaotic, sometimes leading to ejection of one of the bodies. The reason we can even try to model things like galaxies is that the stars are well outside the tidal spheres of the other stars in the galaxy. In this limit, we treat the star as a test particle moving through a time-steady gravitational potential.

Sure, but can we consider this in the context of galaxies? That is can we ignore chaos and tides for the moment and consider our system to be a sort of simplified galaxy? Feel free to change the parameters to meet your needs if required, just clearly express them. Maybe we can move our two mini suns to a farther orbit?

 

[SpaceTiger]

Sure, but can we consider this in the context of galaxies? That is can we ignore chaos and tides for the moment and consider our system to be a sort of simplified galaxy? Feel free to change the parameters to meet your needs if required, just clearly express them. Maybe we can move our two mini suns to a farther orbit?

When modelling galaxies analytically, we usually treat the stars as orbiting in a time-independent potential well that is continuous in space, meaning that we don't treat the stars individually. This potential is generally not spherically symmetric, but for your purposes, it would be far too mathematically cumbersome to attempt such a calculation without spherical symmetry. I set it up in my post here:

https://www.physicsforums.com/showthread.php?t=103450&page=3"

To get the velocity of a circular orbit in a spherically symmetric potential, you need only set the gravitational acceleration equal to the centripetal acceleration:

$$\frac{dV}{dr}={v^2}{r}$$

$$v=\sqrt{r\frac{dV}{dr}}$$

Note that I'm defining V to be the gravitational potential (which has units of energy per unit mass) and it can include contributions from matter, dark matter, and dark energy.

 

[ubavontuba]

That's jumping ahead of where I want to be. Let's back up to the binary system and work from there.

Considering our binary system: As you've noted, the orbital period is greater for this system than the sun/Venus system due to the increased total gravitational potential, right?

So let's consider two much smaller masses (say Jupiter sized) in the same orbits. We'd expect them to be orbiting each other much slower than two solar masses, right?

 

[SpaceTiger]

Considering our binary system: As you've noted, the orbital period is greater for this system than the sun/Venus system due to the increased total gravitational potential, right?

That's right.

So let's consider two much smaller masses (say Jupiter sized) in the same orbits. We'd expect them to be orbiting each other much slower than two solar masses, right?

If they had the same separation, then that's also correct. You can see this from Kepler's Third Law that I quoted in my first post. As mass increases, period decreases.

 

[Chronos]

This is important in accretion models of planetary formation. Big chunks speed up, accreting less massive particles in their path.

 

[ubavontuba]

This is important in accretion models of planetary formation. Big chunks speed up, accreting less massive particles in their path.

Although this leads into an aside that I hadn't intended to pursue, I think this is a very intereting point. Basically, they speed up and catch more loose particles thus increasing their mass allowing them to speed up and catch more loose particles... until eventually all of the loose particles are gone, right?

Are there any hypothesis then as to why the asteroid belt seems so stable? Why is it there? Why don't the bigger chunks catch the smaller chunks? Is it an exploded planet, or leftovers from the formation of the solar system? If the former, that musta' been one heck of a wallop!

 

[SpaceTiger]

Basically, they speed up and catch more loose particles thus increasing their mass allowing them to speed up and catch more loose particles... until eventually all of the loose particles are gone, right?

That's basically right, I think, though many of the loose particles can be expelled by gravitational interactions or radiation pressure (if they're very tiny).

Are there any hypothesis then as to why the asteroid belt seems so stable? Why is it there? Why don't the bigger chunks catch the smaller chunks?

In some cases, they do. See, for example, the case of Ida and Dactyl:

http://www.solarviews.com/eng/ida.htm"

For the most part, though, the asteroids are far enough apart from one another that their motions are dominated by the sun's gravity.

Is it an exploded planet, or leftovers from the formation of the solar system? If the former, that musta' been one heck of a wallop!

There is no one definite answer for the origin of the asteroid belt, but Jupiter likely had something to do with it. See here for a description of some of the theories:

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WGF-458NDKC-24&_coverDate=10%2F31%2F2001&_alid=362473817&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=6821&_sort=d&view=c&_acct=C000051401&_version=1&_urlVersion=0&_userid=1082852&md5=adc29cb2e3865c8a690c5105f405ac8a"

 

[ubavontuba]

SpaceTiger,

Thanks. Those were cool references. I especially liked the photos of Ida and Dactyl. Although it seems to me like the hypothesis proposed in that paper is based on a lot of assumptions, I guess it's as good as any explanation. My favorite unofficial (r.e. "crackpot") hypothesis concerning the asteroids is that it was a planet eaten from within by one or more mini black holes.

 

[SpaceTiger]

Although it seems to me like the hypothesis proposed in that paper is based on a lot of assumptions, I guess it's as good as any explanation.

I think the paper is making it appropriately clear that the issue is not settled. The chaotic behavior of the solar system makes it really hard to obtain any definitive answers about its detailed history.

 

[Chronos]

SpaceTiger,

Thanks . . . My favorite unofficial (r.e. "crackpot") hypothesis concerning the asteroids is that it was a planet eaten from within by one or more mini black holes.

That is unlikely. More like a planet that never formed.

 

[ubavontuba]

That is unlikely. More like a planet that never formed.

Sure, but doesn't it give you goosebumps?

Seriously though, if the asteroids were a "planet that never formed," Why isn't it in the form of a ring that completely encircles the sun?

Is the asteroids' orbital period the same as Jupiter's?

Assuming not, I can see that Jupiter might temporarily both slow their orbital rate and increase their distance from the sun, thus causing them to require enough distance between them to prevent collisions during this tidal effect, but how did they clump to begin with then? And again, why don't they form a full ring? This effect shouldn't prevent them from spreading out in their orbits, should it?

 

[tony873004]

Seriously though, if the asteroids were a "planet that never formed," Why isn't it in the form of a ring that completely encircles the sun?

Pretty much, it is. There's just a lot of space between the asteroids, so it doesn't look like a ring. But there are 10s of thousands of them (maybe 100s of thousands?) They form a ring.

Is the asteroids' orbital period the same as Jupiter's?

Over long periods of time, the period of trojan asteroids average Jupiter's period. Main belt asteriods have shorter periods.

Assuming not, I can see that Jupiter might temporarily both slow their orbital rate and increase their distance from the sun, thus causing them to require enough distance between them to prevent collisions during this tidal effect, but how did they clump to begin with then? And again, why don't they form a full ring? This effect shouldn't prevent them from spreading out in their orbits, should it?

They do form a full ring. It's just too sparce to look like one. Jupiter won't do that unless they're in one of its resonance zones. Then it will clear a gap. But it's not even a real gap. Its just a gap in a table of data. There's pleanty of asteroids in the "gaps". But they're just passing through. Their SMAs don't reside there. They may have clumped together to begin with when Jupiter was not massive enough to disrupt them (Just a guess). Then Jupiter (and to a lesser extent, Mars) scrambled their eccentricities so they would never have low velocities collisions again, hence, they would tend to break apart upon collision rather than merge. And the collisions would create lots of dust that gets blown away by the solar wind and solar pressure. So that's why the belt doesn't contain much mass.

 

[ubavontuba]

They do form a full ring. It's just too sparce to look like one. Jupiter won't do that unless they're in one of its resonance zones. Then it will clear a gap. But it's not even a real gap. Its just a gap in a table of data. There's pleanty of asteroids in the "gaps". But they're just passing through. Their SMAs don't reside there. They may have clumped together to begin with when Jupiter was not massive enough to disrupt them (Just a guess). Then Jupiter (and to a lesser extent, Mars) scrambled their eccentricities so they would never have low velocities collisions again, hence, they would tend to break apart upon collision rather than merge. And the collisions would create lots of dust that gets blown away by the solar wind and solar pressure. So that's why the belt doesn't contain much mass.

Ah, now I see why there's no defined ring. I hadn't considered that the solar wind could drive away all of the dust and fine debri. It makes sense that their orbits may have been lifted high enough that Jupiter was able to gobble them up. Kind of a cosmic broom (so to speak). Thanks for the explanation.

What is the average density of the asteroid belt? I.e. what is the average distance between asteroids? Are some regions more dense than others?

 

[tony873004]

Ah, now I see why there's no defined ring. I hadn't considered that the solar wind could drive away all of the dust and fine debri. It makes sense that their orbits may have been lifted high enough that Jupiter was able to gobble them up. Kind of a cosmic broom (so to speak).

What is the average density of the asteroid belt? I.e. what is the average distance between asteroids? Are some regions more dense than others?

Well, some asteroids orbit other asteroids. That specific region would be denser. I imagine the inner and outer boundaries are less dense than the middle.

Jupiter could gobble them up, or send them on escape (from the solar system) trajectories, or on collision trajectories with the other planets or the Sun, or capture them into resonances (with the help of a non-jovian influence).

Average density? You'd want to know the total mass of the asteroid belt. Then figure out the area of a circle of radius=outer edge of the asteriod - area of circle of radius=inner edge and compute the density from there. Average spacing... You'd need to know how many asteroids there were. Sounds like a job for Google (or Wikipedia!)

btw... Jupiter is the cosmic broom that keeps Earth from getting pelted as often as it otherwise would.

 

[ubavontuba]

Back to the original subject...

Let's move our orbital model to a larger scale. Let's make it a galaxy.

In this scale, we can have relatively equal masses orbiting a dense core at equidistant orbits, right (no interferring tidal forces?)?

 

[tony873004]

A galaxy isn't exactly like a solar system. In a solar system, (at least in ours), the central mass (Sun) contains more than 99.9% of the mass, and objects orbit it.

By the core of the galaxy, do you mean the central black hole, or it plus the dense concentration of stars that surround it?

In any case, the core, even though it contains a lot of mass, does not contain a significant percentage of the galaxy's mass.

However, the period of the Sun's orbit around the galaxy is the same as it would be if all the mass of the galaxy interior to the Sun were concentrated in the middle.

Unlike solar systems, as you move further from the core, the mass interior to you goes up. So objects orbiting galaxies tend to have the same orbital velocity no matter where they are, at least once you've distanced yourself from the core. It's not a perfect flatline graph, but close. Distant objects still have higher periods since they have longer to travel while going the same speed.

 

[ubavontuba]

By the core of the galaxy, do you mean the central black hole, or it plus the dense concentration of stars that surround it?

I mean to create a simplified thought model of a galaxy. Basically a solid core being orbited by two distant bodies. Is this a satisfactory scenario?

However, the period of the Sun's orbit around the galaxy is the same as it would be if all the mass of the galaxy interior to the Sun were concentrated in the middle.

I thought the reason dark matter was perceived was because this isn't the case. Aren't the orbits supposed to be too fast?

Unlike solar systems, as you move further from the core, the mass interior to you goes up. So objects orbiting galaxies tend to have the same orbital velocity no matter where they are, at least once you've distanced yourself from the core. It's not a perfect flatline graph, but close. Distant objects still have higher periods since they have longer to travel while going the same speed.

Isn't this why we get stable stuctures like spiral arms? However, again aren't the orbits too fast?

 

[tony873004]

In the simplified model, with a solid core and 2 distant bodies, if the 2 bodies are equidistant, they need to be trapped in each other's Lagrange points, or they will ultimately crash into each other, or exchange orbital energy and one ejects the other into intergalactic space, or sends the other one crashing onto the solid core.

Realistically, the time frame for the scenerios I describe would be magnitudes greater than the age of the universe.

mass of the galaxy interior to the Sun =
mass we can see or infer +
mass we can't see or infer.

The orbits are too fast to be orbiting what we can see. The orbits are also too fast to be orbiting what we can see + what we expect exists. This is why this is why dark matter is an interesting topic.