Missing Matter Problem and Galactic Flows

[Michael Faraday]

The standard example of the Missing Mass Problem comes from the rotational profiles of galaxies. By counting up the visible matter, we extrapolate a mass profile for a galaxy. We then apply Kepler's laws (the enclosed mass of a stable orbit can be modeled as a point mass) to calculate the expected velocity:

$f(v) = \sqrt{\frac{GM}{r}}$
​

Where G is the gravitational constant, M is the enclosed mass of an elliptical orbit, r is the radius of the orbit. But this formula assumes a closed, elliptical orbit. I'm sure the data exists, but I haven't been able to find it. How do we know that the Earth, for instance, is not falling towards or away from the center of the Milky Way. That is, when we apply the rules of Keplar's orbits, what information do we have that the orbits of the observed galactic bodies describe a closed ellipse and not a spiral in or out (which would change the amount of missing mass considerably)?

 

[Drakkith]

How do we know that the Earth, for instance, is not falling towards or away from the center of the Milky Way.

We can measure our velocity relative to other stars in the galaxy. We happen to be sitting right in the middle, faster than some and slower than others. Our relative velocity is taking us around the galaxy, not towards or away from the center. We do drift around some thanks to interactions with nearby stars, but overall we're in a stable orbit around the center. Also, orbits do not spiral inward unless the orbiting body has a way of losing kinetic energy.

 

[Michael Faraday]

We can measure our velocity relative to other stars in the galaxy. We happen to be sitting right in the middle, faster than some and slower than others. Our relative velocity is taking us around the galaxy, not towards or away from the center. We do drift around some thanks to interactions with nearby stars, but overall we're in a stable orbit around the center.

The tangential velocity is well understood and measured (I've found numerous papers on the subject) at roughly 235 km/s. What appears to be less well understood is the radial component of that velocity. You say we're in a stable orbit, but I can't find any definitive papers on the subject. Would you mind sending me the reference that you're using?

Also, orbits do not spiral inward unless the orbiting body has a way of losing kinetic energy.

Yes, and stars are supposed to get slower as you get further away from the galactic center, but they don't. If the disk is the result of a galactic collision, then the disk could be the result of the ejecta and be spiraling out or in. I'm looking for a kinematic study that supports the assumption of a closed, elliptical orbit.

 

[PeterDonis]

I'm looking for a kinematic study that supports the assumption of a closed, elliptical orbit.

The orbital period of such an orbit, at our distance from galactic center, is about 250 million years; that's the relevant dynamic time scale. If the solar system were not in such an orbit, it would not have remained stable for a time 18 times longer than that time scale.

 

[Michael Faraday]

The orbital period of such an orbit, at our distance from galactic center, is about 250 million years; that's the relevant dynamic time scale. If the solar system were not in such an orbit, it would not have remained stable for a time 18 times longer than that time scale.

Whoa! That's the part I'm missing. How do you know we've been stable for 18 times 250 million years? We could have been spiraling outward (at roughly 0.5 km/s by my calculations) or spiraling in (again, as ejecta as the result of a collision). Would you please point me in the direction of the reference you use?

 

[Chalnoth]

Whoa! That's the part I'm missing. How do you know we've been stable for 18 times 250 million years? We could have been spiraling outward (at roughly 0.5 km/s by my calculations) or spiraling in (again, as ejecta as the result of a collision). Would you please point me in the direction of the reference you use?

This is largely down to the way gravity works.

Gravity, in the Newtonian approximation (which is good enough for the orbits of most stars in our galaxy), is a force that conserves energy. For an orbit to spiral inward or outward would require a loss or gain of energy. So the question arises: where would the energy be coming from or going to?

A close pass between our Sun and another star might transfer some energy between the two stars, but stars rarely pass that close to one another. For the most part, the Sun's orbital energy remains pretty constant. There's a little bit of friction from the interstellar gas, but that's a pretty small effect.

 

[Michael Faraday]

This is largely down to the way gravity works.

Gravity, in the Newtonian approximation (which is good enough for the orbits of most stars in our galaxy), is a force that conserves energy. For an orbit to spiral inward or outward would require a loss or gain of energy. So the question arises: where would the energy be coming from or going to?

A close pass between our Sun and another star might transfer some energy between the two stars, but stars rarely pass that close to one another. For the most part, the Sun's orbital energy remains pretty constant. There's a little bit of friction from the interstellar gas, but that's a pretty small effect.

Where is the energy lost or gained: Collision with a dwarf galaxy could either add or remove energy/mass from the Milky Way. For all we know, our local group of stars could be the remnants of a galaxy that was merged billions of years ago. Most spirals are believed to have been formed from mergers. There's substantial evidence of large streams of matter in our halo from previous collisions. So can we move on to the question of whether we're falling in, falling out or in a stable orbit?

 

[Chalnoth]

Where is the energy lost or gained: Collision with a dwarf galaxy could either add or remove energy/mass from the Milky Way. For all we know, our local group of stars could be the remnants of a galaxy that was merged billions of years ago. Most spirals are believed to have been formed from mergers. There's substantial evidence of large streams of matter in our halo from previous collisions. So can we move on to the question of whether we're falling in, falling out or in a stable orbit?

The energy transfer only occurs during the collision. The collision (or close pass) changes the orbit. The orbit immediately after the interaction that changes the energy of the orbit is a stable orbit (until a new interaction occurs).

 

[Michael Faraday]

The energy transfer only occurs during the collision. The collision (or close pass) changes the orbit. The orbit immediately after the interaction that changes the energy of the orbit is a stable orbit (until a new interaction occurs).

I can't figure out what you are talking about. When exactly did the collision between the proto-galaxies stop?
(OK, the moon was a bad example, but the collision between proto-galaxies will be felt for hundreds of millions, perhaps billions of years. There will be ebbs and flows and ejecta from the collision).

 

[Drakkith]

I can't figure out what you are talking about. When exactly did the collision stop?

If two galaxies collide, the collision is over with once the two galaxies are separated again. If they don't separate, then the collision just rolls into a complicated merging process. Collisions of galaxies are more like long-term processes anyways, not single events, since the relevant interactions take place over the course of hundreds of millions of years.

The moon spirals away from Earth at roughly 2 meters every century. Are you taking the position that there's some external, undiscovered force driving it away from Earth?

The Moon gets further away because of the transfer of rotational energy from the Earth to the Moon. This slows down the Earth's rotation and accelerates the Moon, resulting in a larger orbit.
http://en.wikipedia.org/wiki/Tidal_acceleration

 

[Chalnoth]

I can't figure out what you are talking about. When exactly did the collision between the proto-galaxies stop? The moon spirals away from Earth at roughly 4 meters every century. Are you taking the position that the energy transfer has stopped because the collision is over and there's some external, undiscovered force driving the moon away from Earth?

That's a very different effect. The Moon's orbit is slowly increasing because of tidal effects: the fact that the Earth's rotational period is different from the Moon's orbital period transfers energy from the Earth's rotation to the Moon's orbit. This effect is only significant for objects that are very close to one another.

The solar system itself is more than 3 light years from any other star. There just aren't any collisions/close passes going on at the moment. Nor are there likely to be any time soon.

 

[Michael Faraday]

If two galaxies collide, the collision is over with once the two galaxies are separated again. If they don't separate, then the collision just rolls into a complicated merging process. Collisions of galaxies are more like long-term processes anyways, not single events, since the relevant interactions take place over the course of hundreds of millions of years.

Where is the energy lost or gained: Collision with a dwarf galaxy could either add or remove energy/mass from the Milky Way. For all we know, our local group of stars could be the remnants of a galaxy that was merged billions of years ago. Most spirals are believed to have been formed from mergers. There's substantial evidence of large streams of matter in our halo from previous collisions. If you've ever watched a simulation of galaxy collisions, it's not immediately obvious what parts fall back in and what parts are ejected (into the halo). Can we move on to the question of whether we're falling in, falling out or in a stable orbit, please?

 

[Chalnoth]

That question has been answered, a few times. Our solar system's orbit is relatively stable until the Sun experiences a close pass with another star.

 

[Michael Faraday]

That question has been answered, a few times. Our solar system's orbit is relatively stable until the Sun experiences a close pass with another star.

You've expressed your opinion. I asked for references. I would like to read up on the data used and the method employed to determine that the orbit is closed. I'm assuming you must have read something also to have such a strong belief in the subject, so please share with me how you came upon this information.

 

[Bandersnatch]

For an orbit not to be closed the mass inside the radius of the orbit would need to be changing. With constant central mass all orbits are closed. For the proof of this statement look towards Newtons Principia, or read on central force motion.
Galactic mergers can only be relevant while in progress, as has been mentioned already. You can't have a spiral for an orbit if there's no mass currently being added. Past mergers affected past orbits. Current orbits have to be stable closed ones (perturbations notwithstanding), as there are no on-going mergers.

 

[Drakkith]

You've expressed your opinion. I asked for references. I would like to read up on the data used and the method employed to determine that the orbit is closed. I'm assuming you must have read something also to have such a strong belief in the subject, so please share with me how you came upon this information.

Here you are: Radial Migration of the Sun in the Milky Way: a Statistical Study

ABSTRACT The determination of the birth radius of the Sun is important to understand the evolution and consequent disruption of the Sun’s birth cluster in the Galaxy. Motivated by this fact, we study the motion of the Sun in the Milky Way during the last 4.6 Gyr in order to find its birth radius. We carried out orbit integrations backward in time using an analytical model of the Galaxy which includes the contribution of spiral arms and a central bar. We took into account the uncertainty in the parameters of the Milky Way potential as well as the uncertainty in the present day position and velocity of the Sun. We find that in general the Sun has not migrated from its birth place to its current position in the Galaxy (R). However, significant radial migration of the Sun is possible when: 1) The 2 : 1 Outer Lindblad resonance of the bar is separated from the corrotation resonance of spiral arms by a distance ∼ 1 kpc. 2) When these two resonances are at the same Galactocentric position and further than the solar radius. In both cases the migration of the Sun is from outer regions of the Galactic disk to R, placing the Sun’s birth radius at around 11 kpc. We find that in general it is unlikely that the Sun has migrated significantly from the inner regions of the Galactic disk to R.

 

[Tanelorn]

Michael, have you seen these pages, they may help some:

http://en.wikipedia.org/wiki/Spiral_galaxy#Spiral_arms
http://en.wikipedia.org/wiki/Gravitationally_aligned_orbits

 

[Chronos]

The Hulse-Taylor study closed the book on kinetic energy loss due to gravitational waves, as predicted by Einstein. It appears consistent with the figure offered by PeterDonis.

 

[Michael Faraday]

Here you are: Radial Migration of the Sun in the Milky Way: a Statistical Study

Correct me if I'm wrong, but this paper appears to be the results of a simulation, not a kinematic study. Am I missing something?

 

[Michael Faraday]

The energy transfer only occurs during the collision. The collision (or close pass) changes the orbit. The orbit immediately after the interaction that changes the energy of the orbit is a stable orbit (until a new interaction occurs).

Watch any simulation of galaxies colliding and you'll see streams of ejecta. It's apparently fairly common. Note this simulation:



Satellite galaxies are formed from the collision. Apparently there's considerable evidence that our satellites may have formed the same way (with outflows from a collision). I'm not saying it is, I'm not saying it isn't, but I'd like to know how we know that we aren't currently part of the outflow of one of these jets. Alternatively, you may have some information that we've not collided with any other galaxies in the last 12 billion years and thus can use our Keplarian assumptions about orbits.

 

[Drakkith]

Correct me if I'm wrong, but this paper appears to be the results of a simulation, not a kinematic study. Am I missing something?

A kinematic study is a study of the collective motion of of stellar objects. That is exactly what this paper does. Of course it uses a simulation. All studies of motion use simulations. A table of stellar velocities is exactly that. A table of data. It's meaningless without a model or simulation to use that data.

Satellite galaxies are formed from the collision. Apparently there's considerable evidence that our satellites may have formed the same way (with outflows from a collision). I'm not saying it is, I'm not saying it isn't, but I'd like to know how we know that we aren't currently part of the outflow of one of these jets.

Several reasons. For one, the milky way is not currently undergoing a merger like the simulation you linked shows. We don't have huge streams of stars and gas and another disrupted major galaxy nearby. (The LMC and SMC have been disrupted by the milky way, but this is not like the merger process your video shows) Two, we know our solar system's velocity through space relative to the rest of the galaxy, and it isn't taking us out of the disk.

 

[Drakkith]

Apparently there's considerable evidence that our satellites may have formed the same way (with outflows from a collision).

A quick search for the origin of the LMC and SMC does not support this. Please provide a reference for this claim.

 

[Michael Faraday]

A quick search for the origin of the LMC and SMC does not support this. Please provide a reference for this claim.

http://www.swinburne.edu.au/media-c...lenge-standard-model-of-galaxy-formation.html

Two, we know our solar system's velocity through space relative to the rest of the galaxy, and it isn't taking us out of the disk.

arxiv.org/abs/1209.0759
We apparently are falling in at a rate of roughly 10 km/s. You seem unusually misinformed for a mentor. I have a tough time believing that you don't know the difference between a simulation and a kinematic study. Please, I'd prefer to not get any responses at all than misinformation.

 

[Drakkith]

We apparently are falling in at a rate of roughly 10 km/s.

Which isn't taking us out of the disk, like we've said. 10 km/s is well within the limits to stay within the disk.

You seem unusually misinformed for a mentor.

Mentors aren't experts (well, I'm not at least). We're moderators. We just make sure people follow the rules.

 

[Bandersnatch]

arxiv.org/abs/1209.0759

You have just linked to a paper that analyses the velocities of 3k+ stars from 4 to 14 kpc from the galactic centre as collected by APOGEE. This is nearly half of the radial extent of the Galaxy showing a flat velocity curve. The analysis averages out peculiar velocities, since these are always going to be present due to local interactions and you need a global behaviour.
This is consistent with a stable, uniformly rotating disc.

Type in 'milky way structure' into arxiv's title search and you'll get swamped by papers analysing just that using a variety of methods.

No mass outflows are observed. Neither is there evidence of recent collisions with other galaxies.

The lack of ongoing mergers means that mass can be treated as constant. While the MW as pretty much every other galaxy out there did form via mergers with other galaxies (example simulation here:

you seem to hold a belief that a past merger, already incorporated into the structure, somehow invalidates the mass constancy assumption. Please explain why do you think it is so.

 

[Michael Faraday]

you seem to hold a belief that a past merger, already incorporated into the structure, somehow invalidates the mass constancy assumption. Please explain why do you think it is so.

Beautiful simulation. Thanks for the link. Here's another showing the (relatively) recent collision with the Sagittarius Dwarf galaxy:



The entire article can be found here:
http://astronomynow.com/news/n1109/21milkyway/
I'm not saying I believe in inflows, outflows or a closed orbit. I'm looking for people who've studied the issue without making Keplarian assumptions about mass and orbits.

 

[Bandersnatch]

Have you read the paper you linked to then? Or typed 'milky way structure' into arxiv title search and read any of those?

 

[Michael Faraday]

you seem to hold a belief that a past merger, already incorporated into the structure, somehow invalidates the mass constancy assumption. Please explain why do you think it is so.

There you go again, using that word: assumption. The Earth moves away from the sun every century by 3.8 meters because the sun is constantly loosing mass. If something as small as the solar system loses mass, then why would you think something as complex as the center of our galaxy would have a constant mass? I've studied cosmology for some time and have yet to run across the Principal of Constant Mass in galaxy formations.

In the space of 1/2 hour I found half-a-dozen articles on the past collisions with Sag. DET. including a Wiki page. While there's some healthy discussion over how many times it's collided and how much it's contributed to the spiral shape of the Milky Way, there's no debate that we've collided with it at least once in the last billion years.

 

[PeterDonis]

The Earth moves away from the sun every century by 3.8 meters because the sun is constantly loosing mass. If something as small as the solar system loses mass, then why would you think something as complex as the center of our galaxy would have a constant mass?

And despite this, the Earth has not escaped from the solar system in the last 4.6 billion years. In other words, this small mass loss rate has no impact on the overall stability of the solar system. And the stability of the system is the topic under discussion, not whether or not the mass is exactly constant. If the known rate of mass loss does not impact the stability of the solar system, why would you expect mass loss to impact the stability of a much larger system, the entire Milky Way galaxy?

 

[Michael Faraday]

And despite this, the Earth has not escaped from the solar system in the last 4.6 billion years. In other words, this small mass loss rate has no impact on the overall stability of the solar system. And the stability of the system is the topic under discussion, not whether or not the mass is exactly constant. If the known rate of mass loss does not impact the stability of the solar system, why would you expect mass loss to impact the stability of a much larger system, the entire Milky Way galaxy?

Don't tell me what the topic is, I started the thread. I want to know where the data is to show whether we're falling in, falling out or in a stable orbit. Thanks for the answer, but it's not a question I've asked.

It turns out we're falling in by 10 km/s according to one study, but I had to go to Stack Exchange to find the answer. I'm taken back by the presumptions and assumptions by people labelled 'Mentor's on this board.

 

[PeterDonis]

I want to know where the data is to show whether we're falling in, falling out or in a stable orbit.

If your definition of "stable" is "unchanging", that seems like a much too restrictive definition. The Earth's orbit about the Sun is stable--it's lasted for 4.6 billion years--but it is not unchanging, as you pointed out. Similarly, even if the solar system is "falling in" towards the center of the galaxy at 10 km/s, that does not mean the solar system's orbit is not stable, only that it's not unchanging. (Actually it doesn't necessarily even mean that; an unchanging elliptical orbit has nonzero radial velocity everywhere except at the points of minimum and maximum distance from the central object.) Expecting unchanging orbits for anything in the real universe where there are always perturbations from other bodies is unrealistic, so defining "stable" as "unchanging" basically means no orbit is ever stable.

 

[Michael Faraday]

If your definition of "stable" is "unchanging", that seems like a much too restrictive definition. The Earth's orbit about the Sun is stable--it's lasted for 4.6 billion years--but it is not unchanging, as you pointed out. Similarly, even if the solar system is "falling in" towards the center of the galaxy at 10 km/s, that does not mean the solar system's orbit is not stable, only that it's not unchanging. (Actually it doesn't necessarily even mean that; an unchanging elliptical orbit has nonzero radial velocity everywhere except at the points of minimum and maximum distance from the central object.) Expecting unchanging orbits for anything in the real universe where there are always perturbations from other bodies is unrealistic, so defining "stable" as "unchanging" basically means no orbit is ever stable.

I'm not the one who came up with these laws. The assumption of missing mass is based on Kepler's 3rd law which requires a closed orbit (and a constant amount of mass inside the closed orbit). If the orbit isn't closed and/or the mass isn't constant, then the assumption of Kepler's 3rd law needs to be examined. That's what I'm doing with these questions.

 

[Drakkith]

It turns out we're falling in by 10 km/s according to one study, but I had to go to Stack Exchange to find the answer.

That's because you never asked whether we had a radial velocity. Your question has been about whether or not we are spiraling in or out. All the answers have addressed this question, and the answer is that we are not spiraling in or out, despite the fact that we have a radial component to our velocity. As PeterDonis said, having a radial velocity of -10 km/s does not mean that we are spiraling inwards towards the center of the galaxy, it just means our orbit is not perfectly circular.

I'm not the one who came up with these laws. The assumption of missing mass is based on Kepler's 3rd law which requires a closed orbit (and a constant amount of mass inside the closed orbit). If the orbit isn't closed and/or the mass isn't constant, then the assumption of Kepler's 3rd law needs to be examined. That's what I'm doing with these questions.

Going by this and your original post, you seem to think that the discrepancy in the orbital velocity of stars seen in the outer areas of a galaxy can be explained by mass flow? Is that right?

 

[Michael Faraday]

As PeterDonis said, having a radial velocity of -10 km/s does not mean that we are spiraling inwards towards the center of the galaxy, it just means our orbit is not perfectly circular.

The figure quoted in the article, 10 km/s, if true and an average mean, means that in 1 billion years ago we were 10 kpc. further out than we are now. In about 850 million years we will have fallen into the black hole in the center of the galaxy.

That word, spiral, I don't think it means what you think it means.

 

[Bandersnatch]

-10 km/s radial velocity in the galactocentric coordinates is not the same as falling in or spiralling in. Every elliptical orbit will have a radial component everywhere apart from peri- and apoapsis.
Furthermore, it tells you nothing about the kinematics of the galaxy as a whole - you need to compare it with velocities of other stars. It's the bulk motion that defines the galactic structure. The Sun has got -10 km/s radial velocity, other stars have different velocities, including in the opposite direction.