Elysium and Interior Solutions

Tom,

If star clusters were uniformly dense, then my model would predict decreasing velocities toward the center. But their densities are very far from being uniform; mass is highly concentrated near their centers and falls steeply to the background value beyond. So my model also expects velocities to generally increase toward the center for both radial and proper motion velocities.

I'm not aware of any published work that addresses the anomaly uncovered by Drukier et al, in which proper motion velocities are observed to be significantly higher than radial velocities, especially within the inner few parsecs of star clusters. Would you be so kind as to provide a reference or two?


RBenish

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Benish</i>
<br />If star clusters were uniformly dense, then my model would predict decreasing velocities toward the center.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Then it would do the same for a tunnel through the Earth because the Earth also increases density as the center is approached.

So your test boils down to one of how much speed the falling soccer ball will attain -- a Newtonian amount, or a lesser amount. You need to find a way to quantify this so a test becomes possible. Then (only after quantification is complete), apply that test to clusters. If it doesn't work, it would be a violation of "controls against bias" to rework the prediction once it has failed. When scientists are allowed to do that, it becomes practically impossible to falsify any idea, no matter how wrong it may be.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">I'm not aware of any published work that addresses the anomaly uncovered by Drukier et al, in which proper motion velocities are observed to be significantly higher than radial velocities, especially within the inner few parsecs of star clusters. Would you be so kind as to provide a reference or two?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">I'm sure you are already very familiar with the Big Bang's "dark matter" (added as needed to solve any problem) and with Milgrom's MOND (setting a minimum to acceleration). I have not studied the cluster problem and do not know the specifics of how these two models address that particular issue, even though I'm sure I've encountered such articles.

But I can address the Meta Model's expectations. MM predicts a finite range of gravity, and galaxy velocity curves indicate that range is 1-2 kpc, above which the inverse square law transitions to an inverse linear law. As applied to clusters, in Newtonian dynamics, a star's speed is either above or below escape speed and remains that way throughout its orbit unless another star is encountered. However, in MM, escape speed falls off faster than star speed with distance from the center, so escape is easier, and non-escaping stars well away from the cluster center must have lower than Newtonian speeds. If cluster mass is then inferred from these relatively low speeds, the result will be an underestimate of that mass and will make speeds near the center appear anomalously high. That is a quick explanation of the proper motion speed anomaly.

As for radial speeds, another factor enters. In MM, gravitational shielding becomes a factor for the more massive stars. The larger stars really have more mass than conventional stellar evolution models expect. So gravitational redshifts are also more of a factor than the standard model expects because part of a large star's mass is hidden by shielding. And gravitational redshifts are greatest for the more massive, lower velocity stars; and least for the less massive, higher velocity stars. So in both cases, gravitational redshift dispersion adds to velocity dispersion in such a way as to decrease the combined dispersion seen in radial velocities. -|Tom|-

Tom,

Especially when differences in predictions among competing models is small, it is of course best to have a quantitative prediction in the books <i>before</i> observations become robust enough to say which prediction is closest to the truth. But qualitative predictions and roughly emerging patterns in observational data are sometimes all we have. Even this circumstance can be decisive when differences in predictions are not small but rather large. For the purpose of testing my model, the best astronomical observations are unlikely to ever come close to what could be found in an Earthbased laboratory.

Meanwhile, we can discuss the roughly emerging pattern in Globular Cluster velocity dispersions.

Both Big Bang inspired "dark matter" and the Modified Newtonian Dynamics (MOND) of Milgrom (et al) are typically invoked on the scale of galaxies and clusters of galaxies (tens of kiloparsecs to megaparsecs). Since I have been intently looking for any kind of discussion bearing on the radial-vs-proper motion anomaly for some some now, and especially since the anomaly occurs on vastly smaller scales, I would be surprised to learn of such a discussion involving dark matter or MOND. Please let me know if you recall where you saw such a discussion.

As for the Meta Model, I have the following comments:

First, we again have the scale at which the anomaly occurs. If gravity goes from 1/r^2 to 1/r at about 1-2 kpc, that is at least two orders of magnitude larger than the distances characteristic of the cores of Globular Clusters.

Second, even if this effect took place within Globular Clusters, it would be just as evident by radial velocity measurements (Doppler shifts) as by proper motion measurements (plane of the sky position changes). I don't see why velocities measured by one method should be systematically less or greater than velocities measured by the other method.

And third, although Globular Clusters sometimes exhibit "mass segregation" and other star population patterns and peculiarities, any such population is always well mixed with regard to angle with respect to the cluster center; i.e., it would be concentric. Therefore, if certain types of stars were to have intrinsically more red-shifted spectra, if sufficiently large and unaccounted for, this effect could make for a spurious estimate of the whole cluster's velocity with respect to Earth. Conceivably, if this extra redshift is sufficiently large, with <i>lots</i> of really good data it might be ferreted out by revealing spurious recessional movement of the sub-population with respect to the center of the other stars. But, basically, it would enter in as an additive constant (of about the same value for stars with about the same mass, as you have described it) whether the star is moving toward us or away, whether it's on the near side of the cluster or the far side, whether its orbit with respect to the center is circular or radial. I don't see how this could make the average line of sight velocities appear smaller compared to the average proper motion velocities. If anything, if the offset with respect to the center (mentioned above) were detectable, this could possibly make the radial velocity dispersion slightly <i>larger</i>, but not smaller.

If I understand them correctly, the effects you have mentioned are observable in principle, but they would not have the same signature as would be needed to make proper motion velocity dispersions (especially near the clusters' centers) appear larger than line of sight velocity dispersions.

Whereas, if orbits with respect to the clusters' centers tended to be circular rather than radial, that would explain it. In any case, the analysis of the velocity dispersions is a complicated business, and the objects under study are extremely remote. Even if my model were not the only one to qualitatively predict the observed astronomical pattern, for the sake of completeness (if nothing else) it would be advisable to test the kinematic interior solutions with a laboratory experiment.


RBenish

Correction to previous post:

In the paragraph concerning the third point about Meta Model, the first sentence ending, "it would be concentric" should read, "it would be spherically symmetric."

RBenish

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Benish</i>
<br />If gravity goes from 1/r^2 to 1/r at about 1-2 kpc, that is at least two orders of magnitude larger than the distances characteristic of the cores of Globular Clusters.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">The transition is occuring at all scales because it has an exponential character. The half-width of the exponential is 1-2 kpc. But there is still plenty of action at the scale of globulars too. However, if it were as strong as for galaxies, globulars too would be flattened into a plane instead of spherical.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">even if this effect took place within Globular Clusters, it would be just as evident by radial velocity measurements (Doppler shifts) as by proper motion measurements (plane of the sky position changes). I don't see why velocities measured by one method should be systematically less or greater than velocities measured by the other method.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">The meaning of my final paragraph about gravitational shielding must have been obscure. The two effects have different explanations. My time is too tight at the moment to attempt a fresh explanation. But if a re-read doesn't do it, please raise it again in a few days.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">if certain types of stars were to have intrinsically more red-shifted spectra, if sufficiently large and unaccounted for, this effect could make for a spurious estimate of the whole cluster's velocity with respect to Earth.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">That is true. And just such an excess redshift effect is observed but unexplained in the standard model.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">If I understand them correctly, the effects you have mentioned are observable in principle, but they would not have the same signature as would be needed to make proper motion velocity dispersions (especially near the clusters' centers) appear larger than line of sight velocity dispersions.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">It's probably my bad explanation. The model I was trying to describe does have the right signature to explain both effects, but via two different features of PG.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">Whereas, if orbits with respect to the clusters' centers tended to be circular rather than radial, that would explain it.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">How so? Orbits of any eccentricity, including zero, are subjects to the same test. -|Tom|-

Tom,

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">Benish:

If orbits with respect to the clusters' centers tended to be circular rather than radial, that would explain it.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">TVF:

How so? Orbits of any eccentricity, including zero, are subjects of the same test?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote"><hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

This can be understood by considering the extreme cases.

First, let's consider the extreme of all stars in a cluster being on circular orbits whose planes are randomly oriented. Note that the pattern about to be explicated for this extreme case is not controversial with regard to gravity model: All models (to my knowledge) predict the existence of orbits having constant speed at constant distance from the mass center.

Within the group of circularly orbiting stars there are two sub-group extremes: 1) Orbits whose planes contain the line of sight, and 2) orbits whose planes are perpendicular to the line of sight. For all stars in sub-group (1) the maximum proper motion (hereafter, PM) velocities will be when the orbit crosses the visual center of the cluster. For all of these stars crossing the visual center (or nearly crossing the center) the line of sight (hereafter, LOS) velocities, on the other hand, will be zero (or nearly zero). For sub-group (2), orbit planes perpendicular to the line of sight, the PM velocities will all be the maximum that the orbit contains; whereas the LOS velocities will all be zero (or nearly zero).

For this extreme case, then, both sub-groups (1) and (2), LOS velocities will be a maximum at the <i>edges</i> of the orbits, removed from the center (1), or zero (2). A graph that roughly indicates the shape of this pattern may be found at:

[url] www.gravitationlab.com/Extreme-Velocity-Dispersions.html [/url]

Already we see that the pattern that my model predicts begins to emerge from consideration of just the circular orbit extreme. To make sure the other extreme doesn't contain some kind of compensating effect, we turn to it next.

Now consider the extreme of all stars in a cluster being on perfectly radial orbits whose directions are randomly oriented. It needs to be mentioned here that, since PM velocities involve motion in two directions (x and y, say) but LOS velocities involve motion only in one direction (z, say), to make a fair comparison between the respective dispersions, it is standard procedure to include only one PM direction at a time. (This fact lessens the degree of the conclusion reached in the above discussion concerning circular orbits, but the essence of the argument holds true.)

Note that, unlike the case for circular orbits, in the case of radial orbits the velocities of the trajectories are "controversial." My model of gravity deviates strongly from Newton near the clusters' centers, where the density levels off.

First we consider the Newtonian prediction, in which case the velocity of a radial trajectory is a maximum as it crosses the center of the cluster. Consider again two sub-group extremes: 1) Stars moving along, or nearly along, the line of sight, and 2) Stars moving perpendicular, or nearly perpendicular, to the line of sight (nearly along the x-axis, say).

We are especially interested in those stars that are very nearly lined up with the cluster center. Considering such stars from sub-group (1), we can see that, within this narrow angle ("tube") will be included stars with the full range of LOS velocities. We are seeing all of the cluster's stars within the diameter of this viewing tube, and, for all of these stars the PM velocity will be nearly zero. Now consider the same narrow viewing angle (through center) but stars from sub-group (2) moving nearly perpendicularly across it (near the x-axis). This will include stars having the same maximum velocity as stars moving along the line of sight. But there will be fewer of them; we are seeing a smaller portion of the x-axis "tube." Being perpendicular to the line of sight, these stars will have nearly zero LOS velocities. But again, there will be fewer of them.

So, near the visual center of the cluster we have:

LOS Observations: Stars moving along the line of sight -- full range of velocities, highest velocities all included, many stars. And Stars moving perpendicular to the line of sight -- near zero velocities, few stars. And

PM Observations: Stars moving along the line of sight -- near zero velocities, many stars. And Stars moving perpendicular to the line of sight -- maximum velocities included, but few stars.

The magnitude of a velocity <i>dispersion</i> depends less on the existence of a few high velocity stars than on the existence of a large number of stars with velocities that include the high range of velocities. Therefore, although the maximum velocities near the center for individual stars would be the same for both LOS and PM measurements, since more stars with relatively high LOS velocities will be included, and these same stars have nearly zero PM velocities, the LOS velocity dispersion near the center will be greater than the PM velocity dispersion.

We are still within the realm of Newtonian physics. As mentioned earlier and elsewhere, since Globular Clusters are very old objects, they are supposed to be dynamically relaxed so as to have radial orbits thoroughly mixed with circular orbits. The imbalance of velocity dispersions for the extreme cases considered above, therefore cancel each other, so that the PM velocity dispersions are supposed to match the LOS velocity dispersions. This is the basis of the "astrometric" distance measurements, as explained earlier and elsewhere.

Some Globular Clusters exhibit small systematic "rotations," but generally, there is no other reason within Newtonian theory to expect significant deviations from isotropy (radial vs circular orbits). To my knowledge, there have been no attempts to explain the pattern in the data that I have pointed out as being due to a preponderance of circular orbits. Evidently, this would be regarded as a strongly non-Newtonian consequence that will be rejected unless more and other physical facts leave no choice.

Be that as it may, we have yet to bring the consequences of my model into the picture. As mentioned before, in this model a perfectly radial orbit will not go through the center. The model predicts a preponderance of near circular orbits to ensure that these old stable systems do not collapse. At moderate to large distances, radial velocities increase toward the center because the mass is very centrally concentrated; the density falls off steeply. But near the center the density flattens out. It is here that the model predicts small radial velocities.

On the web page referred to near the beginning of this post are two graphs of the extreme cases discussed above. The graph involving circular orbits is the same for Newton as for my model (two curves: LOS and PM velocities). The graph involving radial orbits gives an approximation of the Newtonian pattern and the pattern predicted by my model (four curves: Newton -- LOS and PM, and new model -- LOS and PM).

Although the extreme cases graphed and discussed above help to clarify, I believe, how and why the models have the predictions they do and why they differ, real world star clusters are rather more complicated. To give an idea of how risky it can be to draw conclusions from our observations of these beautiful systems, consider the comments of one of the experts in the field, David Merritt. He is referring to a graph showing three dramatically different curves, representing the case where 1) mass follows light, 2) minimum central density, and 3) minimum total mass:

"The three curves in that figure illustrate three <i>mass</i> distributions that are consistent with the Plummer surface brightness and velocity dispersion profiles. In the potentials corresponding to these very different mass density profiles, one can distribute stars in such a way that their surface density and line-of-sight velocity dispersion profiles are <i>exactly</i> equal to those of an isotropic Plummer model. Thus, the central mass density of a star cluster that looks superficially like a Plummer model is uncertain by at least two orders of magnitude, even given perfect velocity dispersion data (Dejonghe & Merritt 1992)."


RBenish