James: I believe that as a crude first approximation, for spiral galaxies with a prominent "bulge" containing much of the galaxy's mass (assuming of course no DM halo), it is justifiable to treat the galaxy as a point source... but only as a crude first approximation. And if the galaxy has no prominent bulge, this treatment is not acceptable. Moreover, I can confirm that the shell theorem is absolutely not applicable for a disk galaxy; matter outside a spherical shell may have a significant effect on orbits within that shell, to the extent that in many (not at all pathological) cases, stable circular orbits don't even exist! This I can confirm not just from the theory but also from direct experience with simulations.

That said, serious (or at least, what I'd call serious) investigations of galactic velocity distributions do take into account the actual distribution of matter in the galaxy (or at least, some reasonable approximation of it, e.g., by assuming that the galaxy is a thin, rotationally symmetric disk and representing the mass distribution as a function of radial distance).

V. Toth,

Thanks! I generally concur in my opinion, but the rationale for galactic dark matter halos has been based on rotation curve charts and the expectation that the rotational velocity of individual test bodies should diminish as a function of their radial distance. That they do not is the explanation given for the hypothesis that some additional, undetected peripheral mass must exist. In fact, the prescribed solution is typically an enormous elliptical if not spherical distribution of mass for the entire galaxy - thereby conforming at least approximately to shell theorem's specifications!

I'm pretty certain that what is commonly referred to as a thin disk distribution of mass is derived by individually evaluating an array of test objects at varying radial distances to determine the mass required to produce _each_ test object's measured velocity.

However, this is accomplished by determining the mass necessary at a separation distance that is implicitly equal to the test subject's radial distance from the galactic center! As a result, the _amount_of_mass_ required is determined as though it was a _single_point_mass_ - even though it is considered to be distributed... IMO, since the mass is actually _not_ all located at the galactic center, the mass thought to be necessary to produce the observed rotational velocities is _overestimated_ - as an increasing function of radial distance!

Can you refer me to some galactic rotational analysis supporting the requirement for a dark matter halo that does _not_ rely on representing galactic masses as a single point - whose gravitational potential is diminished by the inverse square of a single separation distance (for each test object)?

James, I think a definitive work on this subject is not really a paper but a book: Galactic Dynamics by Binney and Tremaine.

V. Toth,

A specific reference would be helpful. Did you find some rotational analysis that does not rely on collectively representing galactic masses as a singly point whose gravitational potential is diminished by the inverse square of a single separation distance for each test object?

James, for instance Moffat and Rahvar (http://arxiv.org/abs/1306.6383) used an extended disk model in their recent study of galaxies in the context of MOG/STVG gravity. There are also plenty of papers out there that use N-body simulation of galaxy formation/evolution (i.e., very explicitly not treating the galaxy as a whole as a point source.)

I should have been more specific, since the question relates to the application of analytical two-body Newtonian gravitational equations to spiral galaxies. Actually, I was specifically referring to the referenced Binney & Tremaine book.

Modified relativistic gravitation and n-body simulations are not relevant - IMO their unique methods of determining what gravitational effects are imparted at any given radius do not relate to the question.

Does the shell theorem apply only for spherically symmetric

mass distribution ? For a spiral galaxy there is a cylindrical symmetry

about an axis passing through the center of the galaxy and normal to

the plane of the galaxy. In such a case, will the shell theorem not hold ?

In other words, will matter outside the shell influence those within

the shell ?

Biswajoy,

http://en.wikipedia.org/wiki/Shell_theorem states:

"In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body..."

It goes on, stipulating:

"A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its centre."

Specifically, the theorem is used to allow typical discrete astronomical bodies of mass such as stars and planets to be treated as a proper point mass - as specified in Newton's universal law of gravitation, which determines the attractive force between two point masses.

The often noted cylindrical symmetry is I think more correctly a radial symmetry with the planar disk, but at any rate, IMO the shell theorem proof does not properly apply to it. Only if the total mass is effectively located within a single point does inverse square law apply to planetary system-like orbits (effectively unperturbed by other masses). This is fine for planets within the Solar system, as the Sun contains 99.86% of total system mass.

In the Solar system, typically only the Sun's mass is used, along with the subject planet's radial distance, to independently determine rotational (orbital) velocity for each planet.

Re. "... will matter outside the shell influence those within the shell ?" - shell theorem also stipulates:

"If the body is a spherically symmetric shell (i.e., a hollow ball), no net gravitational force is exerted by the shell on any object inside, regardless of the object's location within the shell."

In (improper) simplistic treatments of galactic rotation, the approximated mass of all objects within a subject body's orbit are combined to derive the total mass imparting gravitational force to the subject - at the inverse square of its radial distance. See http://www.astronomynotes.com/ismnotes/s7.htm and its links to http://www.astronomynotes.com/gravappl/s8.htm#A7.2 for example.

Notice that such simplistic treatments indicate that the expected rotation curves diminish as a function of radial distance - just like planets in the solar system. This is simply because the subject disk objects are being treated as though they were planets orbiting the Sun!

The story is clearly described in Newton's Principia. Briefly,

Book 1 Section 12 'Spherical Bodies' describes the Shell theorem. Yes, when looking at forces that vary as 1.r^2, a spherically symmetric distribution can always be represented, outside the sphere, by a central force from a single central mass equal to the sum of all the masses in a sphere.

This section ends with a warning that these results should not be carried over to non-spherical problems without first proving its validity.

Book 1 Section 13, 'Non-Spherical Bodies' introduces the concept of a Centre of Mass, that is only applicable generally to forces that are proportional to 'r'.  This is definitely NOT applicable to '1/r^2' forces from discs. 

To get a very simple example of the problem, assume a galaxy that consists of three identical stars spaced 'r' apart in a straight line. Place a test particle a distance 'r' further out on the same straight line.  The three forces are

'1/1^2 =1',     '1/2^2=1/4'   and     '1.3^2=1/9'

and their total is 1.36.

Now put the three stars in the centre and the force is '3/2^2=3/4=0.75.

That is the 'central' approach only gives about 1/2 what it should.

If you place the test star only about r/10 further out, and repeat the equation, then produces only a very small fraction (~1/100 if I remember rightly) of the gravity.

Modelling the galaxy as a crude sphere instead of a 'cigar', gives similar less dramatic results, and modelling the galaxy as a sphere gives roughly the 'centre of gravity' result.

The comment about the difference between galaxies with and without central black holes, is quoted by Feng and Gallo as making a small difference only.  Search for these authors to get their  latest publications.

The horror story was some Professor who was invited onto the UK television program specifically discussing this matter. He stated categorically that the solar system showed that the shape of the galaxy was irrelevant. The stars are in a disc (roughly) and the centre of mass approach works for them. What he missed was that 98.8% of the mass of the solar system lies in the sun. This meant that the gravitational field from the sun was barely changed by the outer satellites.  With so much mass at the centre, one would expect the results to be within a percent or two of centre of gravity answer and not wildly different as he suggested. 

I have just read the two references form AstronomyNotes that you suggest. The first of these is totally wrong. It is exactly this that causes the the error that leads to people thinking that they need Dark Matter.

They say " if you know the size of an object's orbit and how quickly it is moving in its orbit, you can find the mass of the central object,"  That is only true if the galaxy is fully spherical.  If it is a disc, then they should do what Newton explicitly suggests, and recalculate the integration of the forces assuming a disc instead of a sphere.

For a sphere, Newton proves that the gravitational filed experienced by a star inside a SPHERICAL galaxy, equals the sum of the gravity from the smaller sphere of all the stars with a radius less than the star concerned.  In other words, the stars that lie between the star and the outer reaches of the galaxy cancel out.  Think if the star we are considering. Draw a line to the centre of the galaxy and create a plane passing through the star under consideration and that line. Now only think about the stars whose radius is greater than the star under consideration.

• those on the 'outside' of this plane are pulling the star away from the centre
• however, there are far more there are far more stars on the inside pulling it towards the centre.

Now, because the stars pulling it away from the centre are closer on average than those pulling it towards the centre, their extra force compensates for the numbers. BUT ONLY FOR SPHERICAL GALAXIES, which are rare if they exist at all.

That article then goes and applies the rule to disc galaxies which is totally wrong.

It is articles as grossly wrong as that that cause the apparent need for Dark Matter.

Grrrrrrrrrrrrrrrrrrr!

Reading Newton's Principia is to be recommended. Particularly Book, sections 12 and 13.

Gilbert,

As a lay person with no formal training in physics, I wholeheartedly agree! I cannot survey how physics is commonly taught at elementary levels, but if this example is representative, it might explain why so many physicists are so deeply convinced that Keplerian expectations of spiral galaxy rotational velocity diminishing with radial distance are correct!

Thanks for your nice explanation for the cancellation of gravitational acceleration imparted to a test body by objects at greater radii. That's also important to consider.

As V. Toth mentions in an early comment, modern, detailed evaluations of spiral galaxy gravitation most often use n-body models to evaluate the interactions of discrete objects - not relying on the galactic center for all interactions. As I understand, these models represent a (significantly reduced) number of bodies, of standard mass sizes (determined by, at best, regional luminosity and standard mass/luminosity relations). However, as I understand, these methods, used to model observed mass (without dark matter), also do not produce the Keplerian expectations of diminishing rotation curves - they produce flat rotation curves like those observed(!) While they do generally indicate rotational velocities lower than those observed, this might well be explained by the models' dependence on derived mass/luminosity relations to misrepresent actual the mass distribution - as certainly some significant mass is effectively non-luminous... It is not just greater than expected rotational velocities - but especially the much greater than expected peripheral rotational velocities (that do not diminish with radial distance) that is the rationale for configuring vast, extended halos of dark matter). The earlier Keplerian presumption is now seldom evaluated with more representative models...

While truly spherical galaxies are rare, elliptical galaxies are quite common. However, their stars do not collectively rotate around the center of mass as do self-interacting disk galaxies - stars more generally independently orbit the galactic center. I think this illustrates the fundamental applicability of shell theorem to these more spherically-symmetrical distributions of mass.

In simple, straightforward terms, I suggest that disk galaxies rotate because stars are more strongly gravitationally bound to their neighboring masses - that it is the loosely bound structures within the disk that collectively rotate around their common center of mass. But then, I was not taught to (forcibly) apply two-body gravitational equations to exceedingly complex disk galaxies in my (misspent) youth...

Regarding "... if you know the size of an object's orbit and how quickly it is moving in its orbit, you can find the mass of the central object," please see http://arxiv.org/abs/1108.1629 - this and other, related studies establish that the hundreds of ordinary galactic halo objects - stars, clusters and dwarf galaxies - orbiting the distant periphery of Milky Way do comply with Keplerian diminishment of rotational velocity as a function of radial distance.

This can be explained by scales of distance. At sufficient distance, even a (sufficiently luminous) planar disk structure can appear as a point source. At the vast distances of galactic halo objects, they gravitationally interact with the primary galactic structures collectively as an effective point-mass.

This, then, begs the question: why is it that disk objects do not comply with Keplerian expectations? IMO, the clear and simple answer is that the distant halo objects orbit the primary galactic structures (including the disk) as a discrete object, whereas disk objects primarily interact locally, with their nearest neighbors.

BTW, the above reference states:

3 THE KEPLERIAN ENSEMBLE METHOD

To estimate the lower bound for the MW mass, we assume that external tracers move as test bodies in the host gravitational field of a compact mass distribution. In this case, at sufficiently large radii, a contribution from the higher multipoles of the field should be small compared to the monopole part, and the motion of distant test bodies should be approximately describable in terms of Keplerian orbits. Because point mass formulae can be used as order-of-magnitude estimates of gravitating mass, the compactness assumption does not necessarily reject the possibility that an extended massive dark matter halo may be present. Also, the compactness assumption does not a priori reject the possibility that the spatially extended dark matter halo component may be absent. The latter hypothesis should be tested in accordance with Occam’s razor rule, before one decides to introduce new forms of matter, such as a ubiquitous invisible substance consisting of dark matter particles of unknown nature and, so far, eluding detection in the laboratory by producing no observable non-gravitational effects (e.g., recent null results, LUX Collaboration (2013)).

Also see http://news.sciencemag.org/2013/01/has-milky-way-lost-weight and http://arxiv.org/abs/1205.6203 - section 6.3 is entitled "A low mass, high concentration halo?" It states:

Given that the total mass within 50 kpc is ∼ 4 ×1011M⊙, our results suggest that there may be little mass between 50 and 150 kpc. Is most of the dark matter in our Galaxy therefore highly concentrated in the centre?

IMO, if most of the 'missing mass' is found to be contained within the visible galaxy - as these results suggest - it is much more likely to be represented by ordinary 'dim matter' such as undetected dwarf stars, planets, black holes and cold gasses rather than some unidentified exotic new form of undetectable matter.

Serious work does not assume that the baryonic matter is spherically symmetric and does not use the shell theorem. There have been misguided efforts to explain the rotation curves of spiral galaxies using computational methods and confining the matter to a disc (see for example the many papers, but one idea, by Feng & Gallo). What these efforts show is that you would need a huge amount of dark matter (baryonic or not, it doesn't matter) *in the disc* in order to get the observed rotation curves; the mass to light ratio would have to decrease by an (unexplained) order of magnitude from the inside of galaxies to their visible limits. The mass implied in the disc at the position of the Sun would be 5 times that which can be accounted for by gas, dust and stars and would then also produce velocity dispersions perpendicular to the disc several times larger than those observed.

To conclude. Rotation curves imply a large amount of unseen gravitating matter, whether you assume it is distributed spherically or in a disc; velocity dispersions perpendicular to the disc imply that this "dark matter" is *not* concentrated in the disc. Hence the hypothesis of a pseudo-spherical dark halo, which if it is non-baryonic, would not be expected to follow the distribution of gas and dust in the disc.