Is there a 'gentler' MOND effect in our solar system..?

[Tassel]

Yes, that's the one. G is the same as Newton's G, viz. ~6.67E-11 m^3 kg^-1 s^-2. What I changed was the AUn function, which is a multiplier for distance r, so that at 9.5AU, the AUn=9.5, for example. I did not change the G constant here.

One possible indication of this is by using Milgrom’s MOND’s a = (GM a_o/ r^2)^1/2 and modifying it for our solar system, but to drop Newton’s assumption of a constant universal G, and give it a variable value instead. For example, if we ‘assume’ a variable G at the rate of 1G per 1AU with distance from the Sun (at present unsubstantiated empirically), we get an approximation of the Pioneer Anomaly, as follows

Can you resolve the discrepancy between your two quotes above for me?

Other notes:

1. MOND does apply to our solar system. Well, it would if anything around here were accelerating below the a₀ threshold that MOND specifies. According to MOND, the equation you originally modified only applies when the acceleration an object is undergoing is below a₀. Pioneer is certainly experiencing more acceleration than that, so MOND does not apply.

2. You're adding two additional units to the equation in your original post (AU and meters) and it no longer balances. I suppose you can argue you're just adding some sort of dimensionless multiplier for G to the numerator, but now you're claiming that you're not modifying G so I don't know. Either way, you're adding meters to the denominator.

3. In your original post, you're taking an equation that calculates an acceleration, modifying it to produce the number you want, and then declaring it actually calculates the differential between two accelerations. The whole point of my line of questioning was to coax out a method to calculate acceleration directly, so we could compare the values predicted by this new variable-G-MOND-esque physics and Newtonian physics. You provided one and subtracting the results of the two should have yielded the Pioneer acceleration. However, if you start plugging in values to the equation you gave for acceleration (a = (GM a_o/ r^2)^1/2), you will see that the results are wildly different from Newtonian predictions. This is to be expected, since the predictions of Newtonian physics and MOND are very different for galaxy rotation, which is where MOND is actually meant to apply. So comparing the results of your/MOND's equation to Newton does not result in the Pioneer acceleration. And the differences between the two are not constant so it's not a matter of juggling a₀ to your liking.

I'd say this new equation is shot down.

[nutant gene 71]

Can you resolve the discrepancy between your two quotes above for me?

Other notes:

1. MOND does apply to our solar system. Well, it would if anything around here were accelerating below the a₀ threshold that MOND specifies. According to MOND, the equation you originally modified only applies when the acceleration an object is undergoing is below a₀. Pioneer is certainly experiencing more acceleration than that, so MOND does not apply.

2. You're adding two additional units to the equation in your original post (AU and meters) and it no longer balances. I suppose you can argue you're just adding some sort of dimensionless multiplier for G to the numerator, but now you're claiming that you're not modifying G so I don't know. Either way, you're adding meters to the denominator. (bold mine)

3. In your original post, you're taking an equation that calculates an acceleration, modifying it to produce the number you want, and then declaring it actually calculates the differential between two accelerations. The whole point of my line of questioning was to coax out a method to calculate acceleration directly, so we could compare the values predicted by this new variable-G-MOND-esque physics and Newtonian physics. You provided one and subtracting the results of the two should have yielded the Pioneer acceleration. However, if you start plugging in values to the equation you gave for acceleration (a = (GM a_o/ r^2)^1/2), you will see that the results are wildly different from Newtonian predictions. This is to be expected, since the predictions of Newtonian physics and MOND are very different for galaxy rotation, which is where MOND is actually meant to apply. So comparing the results of your/MOND's equation to Newton does not result in the Pioneer acceleration. And the differences between the two are not constant so it's not a matter of juggling a₀ to your liking.

I'd say this new equation is shot down.

Thanks Tassel, let’s test for dimension analysis. My original post above says:

One possible indication of this is by using Milgrom’s MOND’s a = (GM a_o/ r^2)^1/2 and modifying it for our solar system, but to drop Newton’s assumption of a constant universal G, and give it a variable value instead. For example, if we ‘assume’ a variable G at the rate of 1G per 1AU with distance from the Sun (at present unsubstantiated empirically), we get an approximation of the Pioneer Anomaly, as follows:

-a = (GM a_o/ r^2)^1/2, which becomes modified with 1G per 1AU as:

-a = [G(AUn)M a_o/ r(AUr)]^1/2, where AUn is the number of AU distance, and AUr is the distance r for one AU, so with numbers, for Earth’s orbital:

-a = [(6.67E-11)(1)(1.98E+30)(1.2E-10) / (1.5E+11)(1.5E+11)]^1/2, gives us a value of:

-a = (15.8479E+9 / 2.25E+22)^1/2 = (70.435E-14)^1/2

-a = 8.3934E-7 m/s^2, which is three orders of magnitude greater than Pioneer’s –a = ~8E-10 m/s^2, too far out of ball park.

The same calculation for any distance in AU will yield the same result, i.e., at Saturn’s 9.5AU, where r = 1.429E+12 m, gets nearly same result, viz. –a = 8.38E-7 m/s^2

However, what Milgrom calculated for the outer galaxy flat rotation curves may not be the same as what is operable within the limits of our solar system, so that a ‘gentler’ MOND effect may be the case here, which can be calculated as follows, solving for a_os within our solar system:

-a = = [G(AUn)M a_o/ r(AUr)]^1/2, and plugging in known values for Pioneer Anomaly:

-8E-10 m/s^2 = [(6.67E-11)(1)(1.98E+30)(a_o) / 2.25E+22]^1/2, and solving for our solar system’s a_os we get:

-8E-10 m/s^2 = [13.2066E+19)(a_os) / 2.25E+22 ]^1/2

a_os = 1.0908E-16 m/s^2, for our solar system, which is a far lower, gentler value for our solar system then what was computed for the outer galaxy curves, viz. a_o = 1.2 E-10 m/s^2.

This is the bulk of how a_os, the MOND equivalent for Pioneer Anomaly within our solar system, was derived. I set the Pioneer’s anomalous acceleration towards the Sun, -a, at the measured value: -8E-10 m/s^2. Does this balance in the same way Milgrom’s –a = (GM a_o/ r^2)^1/2? Let’s give it a go:

-a = [(m^3 kg^-1 s^-2)(kg)(m s^-2)/ m^2]^1/2 which reduces to:

-a = (m^4 s^-4/ m^2)^1/2, so we’re left with:

-a = (m^2 s^-4)^1/2 = m s^-2

Now, taking the ‘solar system’ modified version, per mine, where ‘n’ is a dimensionless multiplier:

-a = [(m^3 kg^-1 s^-2)(n kg)(m s^-2)/ m^2]^1/2, which reduces to:

-a = [(m^4 s^-4)(n)/ m^2)]^1/2 = [m^2 s^-4 (n)]^1/2 which is the same as above:

-a = (m s^-2)(n), where ‘n’ is dimensionless.

So, Tassel, I don’t see here what you obviously see here. As you can see, all I did for Milgrom’s MOND for our solar system was include a multiplier ‘n’, not changed the dimensions of the original, so to fit it with the Pioneer Anomaly, as measured empirically.


Would you show your ‘dimension analysis’ to show how yours came out different? Thanks.

[Tassel]

Yup, I missed that you replaced r squared with r * 1 AU in meters. My mistake. I withdraw note #2 in my previous post.

[nutant gene 71]

Yup, I missed that you replaced r squared with r * 1 AU in meters. My mistake. I withdraw note #2 in my previous post.

Thanks Tassel, not a problem, though I truly appreciate your looking into this with some detail. I'm still not sure I got this right, maybe replacing r^2 with an 'abridged' version of r*1AU, all in meters, is not totally crikey either. I don't really know what it means, though the numbers seem to work out into something readable. The dimensionless multiplier 'n' is obviously a 'fudge' factor, relating back to the old variable G idea, though it does not have to be, and it could mean something else. I'm still not sure of what exactly myself, so no harm in challenging it some more. Appreciate your input.

[Utad3]

Aren't the ATM ideas presented by nutant gene 71 and Utad3 quite different? Sure they refer to the Pioneer anomaly, but propose quite incompatible explanations, don't they?

If so, then there should be two ATM threads.

If not, can someone please demonstrate how the two are compatible?

No need for any deriving thread; as I see it, the main point is that there are still open some study avenues that can be explored in order to understand the force of gravity in our solar system; it is crucial that every sound one of them can be studied with equal rigorous measure, that no future doubts subsist.
Also thank you for all the patient you have shown.

Utad3, in Aspden’s paper in the ‘Debate’ he says:
(...)
So whether gravity force potential is nearly instantaneous or at some velocity, whether or not greater than c, is still an unresolved issue, IMO.

I fully appreciate the dedication you have put into your analysis; from my layman point of view, you have shown the required honest open-mind needed to pursuit knowledge. Wishing to you the best success on all your quests in the noble Physics field.

See you.

[nutant gene 71]

No need for any deriving thread; as I see it, the main point is that there are still open some study avenues that can be explored in order to understand the force of gravity in our solar system; it is crucial that every sound one of them can be studied with equal rigorous measure, that no future doubts subsist.
Also thank you for all the patience you have shown.



I fully appreciate the dedication you have put into your analysis; from my layman point of view, you have shown the required honest open-mind needed to pursuit knowledge. Wishing to you the best success on all your quests in the noble Physics field.

See you.

Thanks Utad3, I hope to stay the course on this inquiry, though I myself am not totally satisfied with it until something is proven with actual measurements to validate what MOND appears to say for our solar system. This might be achieved with test for the weak Equivalence of massive matter in the outer solar system, from Jupiter outwards, to see if the 'fudge' factor 'n' included in the above MONDian equation has any real validity. Perhaps someday this will be done.

Another way to approach this is to not adjust a_os down from Milgrom's a_o = 1.2E-10 m/s^2, but wiggle the 'n' value instead, down three orders of magnitude to come up with a much lower value, where it would be about a thousandth of 1G per 1AU hypothesized orginially. I haven't worked it out that way, but it's a thought.

[Tassel]

I hope to stay the course on this inquiry, though I myself am not totally satisfied with it until something is proven with actual measurements to validate what MOND appears to say for our solar system.

Nutant, although I misread your initial equation and was mistaken on the units, I think my other points are still valid. What you posted does not appear to work.

[nutant gene 71]

Addendum to my above:

I recalculated what the 'n' fudge factor would need to be to match up with Milgrom's MOND a_0 = 1.2E-10 m/s^2. Tricky, because of square root, but it works out, if matched up with Pioneer Anomaly (-a = 8E-10 m/s^2), to be AUn = ~0.91E-6, dimensionless.

This would mean, hypothetically, that MOND for our solar system, using Milgrom's calculation, would 'grow' G at the very slow rate of about 10^-6, much lower rate than anticipated from my above, to match up with the Pioneer Anomaly. This implies that the local MOND effect is about 6 orders of magnitude lower than calculated for the outer galaxy rotation curves for our solar system, if true. I suspect that such a slow rate of G increase, if true, would be more in line with possible future observations, more hidden within Newton dynamics at such a low value, than the drastic original growth rate (only hypothetically) of 1G per 1AU. This result is 6 orders of magnitude lower for 'n' fudge factor, if Milgrom's numbers are correct, as applied to the calculated Pioneers acceleration towards the Sun.

Well, it smells right, feels right, but is it right? Only a real test for weak Equivalence out there will know for sure.

[nutant gene 71]

Nutant, although I misread your initial equation and was mistaken on the units, I think my other points are still valid. What you posted does not appear to work.

"Does not appear to work" is not specific enough, Tassel. You'll have to show where this is true, especially given that I just posted how there is a compatability between Milgrom's MOND, as he calculated it, with what the Pioneers had been observed in the outer solar system, within a parameter of 6 orders of magnitude lower than my originally assumed 1G per 1AU. If you find the descrepancy, my hat is off to you. I'd love to find the error too!

[Tassel]

"Does not appear to work" is not specific enough, Tassel.

My other point from above:

If you start plugging in values to the equation you gave for acceleration (a = (GM a_o/ r^2)^1/2), you will see that the results are wildly different from Newtonian predictions. This is to be expected, since the predictions of Newtonian physics and MOND are very different for galaxy rotation, which is where MOND is actually meant to apply. So comparing the results of your/MOND's equation to Newton does not result in the Pioneer acceleration. And the differences between the two are not constant so it's not a matter of juggling a0 to your liking.

The reason that MOND is even on the table as a dark matter alternative is because it only applies to objects feeling a very, very low acceleration (ie: the edges of galaxies). As I say in the above quote, if you start trying to apply MOND to our solar system, where acceleration is greater than the a₀ threshold, it no longer works.

[nutant gene 71]

My other point from above:

The reason that MOND is even on the table as a dark matter alternative is because it only applies to objects feeling a very, very low acceleration (ie: the edges of galaxies). As I say in the above quote, if you start trying to apply MOND to our solar system, where acceleration is greater than the a₀ threshold, it no longer works.

I'm not sure I understand what you have in mind. The Pioneer acceleration towards the Sun is found to be about -a = 8E-10 m/s^2, while Milgrom's MOND has a_0 of about 1.2E-10 m/s^2. Others had noted how similar these two values are, so some mused that there seems to be some sort of MOND like behavior for the Pioneer Anomaly. My calculations are more specific, where to match Pioneers to MOND, there is a more direct relatioship, viz. 1.2E-10 m/s^2 times 0.91E-6 'n' for G matches up with my earlier a_0s = 1.09E-16 m/s^2. Either way, the numbers work out, but whether I am right with the original G growing (fudge factor) of 1G per 1AU for the outer solar system, or Milgrom's effective 10^-6 growth rate for G per 1AU, is still up in the air. Who's right here? Which better explains not only the Pioneers's apparent growth of their inertial-gravitational mass (why they're being pulled back towards the Sun), but also anomalous atmospheres for the gas giants, or Titan's very large atmosphere, or neo-protoplanet Pluto's atmosphere for such a tiny body, or perhaps as yet unresolved cometary anomalies (where they gas out pretty far from the Sun, for example) is all still up in the air. That is, these are anecdotal only until such time that we actually measure for G, or for gravitational-inertial mass Equivalence farther out there in our solar system. What I am suggesting, merely a rought 'back of the envelope' proposal, is that perhaps Newton's assumption that his G is a universal constant may be off the mark. Milgrom thinks so, or it so appears, and I think so too. But which value is right for outer solar system's G variable 'fudge' factor? I really don't know, can't know, not yet. But if you have some specific math to show why you disagree, I'd love to see it. My whole point here is that 'dark matter' does not have to be anything more exotic than (invisible) higher G mass for very distant matter, way out there.

Thanks, I do appreciate your persistence in questioning this. Any math you might have to show this wrong is highly appreciated.

[Iori Fujita]

The oblateness of the Sun is estimated at about 9 millionths which is nearly equal zero, while that of the Earth is about 0.35%. But the Sun is self-rotating at the period of 27 days 6 hours 36 minutes at the equator. Then the centrifugal force there is not zero but 1.8x10-5 of its surface gravity.

I think that there is the equivalent gravity concentration around the equator whose increment is almost equal to the centrifugal force at the equator, to keep the Sun near-perfect round. Instead the solar photosphere is about 1 degree hotter near the poles than it is near mid latitudes according to the NOAO Newsletter 52 December 1997, which means for me that the emission of the Sun shifts subtly to the poles and pushes the gravity to the equator. The gravity concentration makes the light away from the equator. Then the equator looks a little bit darker. The more important is that the gravity concentration makes the solar system stable and many planets' orbital motions stay in one plane. Of course I have to mention about the fact that the axis of the Sun's self-rotation tilts 7 degree against the vertical axis of the solar system plane. This makes an unbalance between the centrifugal force and its surface gravity, which will causes the births of sun spots mainly between latitude 5 degrees and 20 in both the northern and the southern hemisphere.

The extreme phenomenon of a rapidly rotating star is "Gravity Darkening". Vega, located 25 light-years away from Earth, makes a full rotation about its axis once every 12.5 hours, while the Sun takes 27 days for one rotation. Vega, by its speedy rotation, bulges 23 percent fatter at its equator.
A traditional explanation says, "The darkening occurs because the gravity darkening star is colder at its equator than at its poles. Its equatorial bulge diminishes the pull of gravity at the equator, which causes the temperature there to decrease." On the contrary, it can be called a mini spiral galaxy where beams of light are pushed away to the upper or lower side of the equator which then looks cooler and darker than at its poles.

http://www.geocities.jp/imyfujita/galaxy/galaxy01.html
Iori Fujita

[nutant gene 71]

Here is a related story to Pioneer Anomaly, which dove tails into my above: "The exception could be very elliptical comets, but no such variance from expected orbital behaviors had been observed to date." However, as this article points out, there may be reason to suspect that very distant comets, those entering the MOND like regions of the outer solar system, may not be where Newtonian gravity dynamics would predict them to be. (I leave this only as a footnote.)

Lost asteroid clue to Pioneer puzzle
http://www.newscientist.com/article....mg18624984.700


Tassel, re yours: "The reason that MOND is even on the table as a dark matter alternative is because it only applies to objects feeling a very, very low acceleration (ie: the edges of galaxies). As I say in the above quote, if you start trying to apply MOND to our solar system, where acceleration is greater than the a0 threshold, it no longer works." I found references to Beckenstein's theory of squaring Milgrom's MOND with Einstein's GR in these two NewScience articles:

Are There Two Types of Gravity?
http://www.newscientist.com/article....000&print=true

Bekenstein says his theory is consistent with general relativity, gravity and MOND. "The theory reduces to Einstein's theory of gravity at high speeds and accelerations well above a0, to Newtonian gravity at low speeds and accelerations above a0, and to MOND at accelerations below a0."

Gravity theory dispenses with dark matter
http://space.newscientist.com/articl...rk-matter.html

MOND, for example, holds that there are two forms of gravity.

Above a certain acceleration, called a0, objects move according to the conventional form of gravity, whose effects weaken as two bodies move further apart in proportion to the square of distance. But below a0, objects are controlled by another type of gravity that fades more slowly, decreasing linearly with distance
...
Furthermore, the team tested the theory against observations of NASA's 34-year-old Pioneer 10 spacecraft, which appears about 400,000 kilometres away from its expected location in the outer solar system. Brownstein says the theory fits observations of the so-called Pioneer anomaly (see New Scientist feature, 13 things that do not make sense), while MOND cannot address it because Pioneer's acceleration is above a0. (bold mine)

I believe this is what you were referring in yours? However, I am not familiar with Beckenstein's work well enough to comment further at this time. Perhaps you know more and would like to elaborate? Thanks.


Iori, thanks for your comment. I find your idea on the Sun's 'near-perfect spherical form' as relating to its equatorial centrifugal force of 1.8E-5 of its surface gravity interesting. Yours: "I think that there is the equivalent gravity concentration around the equator whose increment is almost equal to the centrifugal force at the equator, to keep the Sun near-perfect round. Instead the solar photosphere is about 1 degree hotter near the poles than it is near mid latitudes according to the NOAO Newsletter 52 December 1997, which means for me that the emission of the Sun shifts subtly to the poles and pushes the gravity to the equator." This is intriguing enough to study further, though at the moment I don't have any numbers to work with to show an 'energy to gravity' relationship to account for why gravity is stronger on the Sun's equator than the poles to account for its near-perfect spherical composition. Per the underlying idea of a 'fudge' factor for G in the outer solar system, far from Sun's hot energy, it could possibly be related to distance from the Sun's hot energy, but at this time, this is still too early to speculate further on it. I'll look at the page your referenced above. Thanks.

[czeslaw]

I'm not sure I understand what you have in mind. The Pioneer acceleration towards the Sun is found to be about -a = 8E-10 m/s^2, while Milgrom's MOND has a_0 of about 1.2E-10 m/s^2. Others had noted how similar these two values are, so some mused that there seems to be some sort of MOND like behavior for the Pioneer Anomaly. My calculations are more specific, where to match Pioneers to MOND, there is a more direct relatioship, viz. 1.2E-10 m/s^2 times 0.91E-6 'n' for G matches up with my earlier a_0s = 1.09E-16 m/s^2. Either way, the numbers work out, but whether I am right with the original G growing (fudge factor) of 1G per 1AU for the outer solar system, or Milgrom's effective 10^-6 growth rate for G per 1AU, is still up in the air. Who's right here? Which better explains not only the Pioneers's apparent growth of their inertial-gravitational mass (why they're being pulled back towards the Sun), but also anomalous atmospheres for the gas giants, or Titan's very large atmosphere, or neo-protoplanet Pluto's atmosphere for such a tiny body, or perhaps as yet unresolved cometary anomalies (where they gas out pretty far from the Sun, for example) is all still up in the air. That is, these are anecdotal only until such time that we actually measure for G, or for gravitational-inertial mass Equivalence farther out there in our solar system. What I am suggesting, merely a rought 'back of the envelope' proposal, is that perhaps Newton's assumption that his G is a universal constant may be off the mark. Milgrom thinks so, or it so appears, and I think so too. But which value is right for outer solar system's G variable 'fudge' factor? I really don't know, can't know, not yet. But if you have some specific math to show why you disagree, I'd love to see it. My whole point here is that 'dark matter' does not have to be anything more exotic than (invisible) higher G mass for very distant matter, way out there.

Thanks, I do appreciate your persistence in questioning this. Any math you might have to show this wrong is highly appreciated.

I have a similar thread about Pioneers and MOND. We may work together on this problem.
In my idea is there in space the Vacuum Energy of the gravitational field as a Dark Matter. This vacuum energy is not uniform in the Universe but its density depends on mass distribution. The average mass density on the Observable Universe is about 10^-26 kg/m^3. The Radius of the Observable Universe is about 10^26 m (it is not a present Radius).
The density of the Vacuum Energy increases when it comes close to Mass Centre. It means the density is inversely proportional to the radius.
Vacuum density d = (10^-26 kg/m^3) x Radius(10^26 m)/radius

It means the Mass of the Galaxy is a sum of its rest mass and its relativistic mass (Vacuum Energy of its gravitational field).

The Mass=M+M(grav)
M(grav)=Volume under orbit x Vacuum Energy density
M(grav)=r^3 x d = r^3 x d x R/r = r^2 x d x R
additional Dark Matter (Vacuum Energy of the gravitational field) acceleration
a=F/m=GMm/mr^2=GM(grav)/r^2=GdR
It means the acceleration is about 10^-10 m/s^2 and depens on the Vacuum Energy of the gravitationa field.
The number is close to Milgrom's MOND but it is not an universal constant. It may vary and depends on the Vacuum Energy distribution.

[John Mendenhall]

Here is a related story to Pioneer Anomaly, which dove tails into my above: "The exception could be very elliptical comets, but no such variance from expected orbital behaviors had been observed to date." However, as this article points out, there may be reason to suspect that very distant comets, those entering the MOND like regions of the outer solar system, may not be where Newtonian gravity dynamics would predict them to be. (I leave this only as a footnote.)

Lost asteroid clue to Pioneer puzzle
http://www.newscientist.com/article....mg18624984.700


Tassel, re yours: "The reason that MOND is even on the table as a dark matter alternative is because it only applies to objects feeling a very, very low acceleration (ie: the edges of galaxies). As I say in the above quote, if you start trying to apply MOND to our solar system, where acceleration is greater than the a0 threshold, it no longer works." I found references to Beckenstein's theory of squaring Milgrom's MOND with Einstein's GR in these two NewScience articles:

Are There Two Types of Gravity?
http://www.newscientist.com/article....000&print=true


Gravity theory dispenses with dark matter
http://space.newscientist.com/articl...rk-matter.html


I believe this is what you were referring in yours? However, I am not familiar with Beckenstein's work well enough to comment further at this time. Perhaps you know more and would like to elaborate? Thanks.


Iori, thanks for your comment. I find your idea on the Sun's 'near-perfect spherical form' as relating to its equatorial centrifugal force of 1.8E-5 of its surface gravity interesting. Yours: "I think that there is the equivalent gravity concentration around the equator whose increment is almost equal to the centrifugal force at the equator, to keep the Sun near-perfect round. Instead the solar photosphere is about 1 degree hotter near the poles than it is near mid latitudes according to the NOAO Newsletter 52 December 1997, which means for me that the emission of the Sun shifts subtly to the poles and pushes the gravity to the equator." This is intriguing enough to study further, though at the moment I don't have any numbers to work with to show an 'energy to gravity' relationship to account for why gravity is stronger on the Sun's equator than the poles to account for its near-perfect spherical composition. Per the underlying idea of a 'fudge' factor for G in the outer solar system, far from Sun's hot energy, it could possibly be related to distance from the Sun's hot energy, but at this time, this is still too early to speculate further on it. I'll look at the page your referenced above. Thanks.

An enjoyable, coherent, and intelligent post! Thanks, Nute!

[Tassel]

I'm not sure I understand what you have in mind.

Nutant, I'm pointing out that the equation you say to use for calculating acceleration does not produce the accelerations we observe for any object, anywhere in the solar system. You said to use this: a = (GM a_o/ r^2)^1/2. It does not work:

Acceleration due to the sun's gravity at 1 AU distance:
Newton: a = 5.93E-03 m/s^2
MOND: a = 8.04E-10 m/s^2

At 5 AU distance:
Newton: a = 2.37E-04
MOND: a = 3.60E-10

etc.

This is why MOND doesn't apply to objects experiencing an acceleration greater than a₀.

I believe this is what you were referring in yours?

I guess. If by "mine" you mean that MOND can't produce the Pioneer acceleration, then yes. But it's not like I'm making some startling breakthrough about MOND. MOND doesn't claim to be able to produce the Pioneer acceleration. Which is why I don't understand why you are trying to use it in this way.

[nutant gene 71]

Thanks John, we're just looking, not yet really a theory, still a spec.

Here's something else I came across, which fits rather well into what the Pioneers are doing, though not MOND specific: Abstract: K12.00007 : Modeling the Pioneer anomaly http://meetings.aps.org/Meeting/APR07/Event/65125

What's intriguing is that the PA may have a negative sign below 1AU, which is not too surprising if there is some sort of G related to distance from the Sun, where at 1AU we're at 1G, by definition. Here's what this short abstract says:

Here, the main arguments used to eliminate DM are refuted and then the anomaly is modeled by application of Newton laws to the observed macroscopic properties of DM. Around a central mass M, the modeling predicts a DM distribution that produces the PA at short distances (R smaller than 188 AU) from a star like the Sun, and a flat rotation curve at sufficiently large distances from the center of a galaxy. Below about 188 AU from the Sun, the modeling predicts that the anomaly may be expressed as PA = 8.3E-8 {\{}[R\^{}(-2)] -- 1{\}} cm (s)\^{}(-2). It shows that the anomaly remains fairly constant down to 5 AU, decreases significantly from 5 AU to 1 AU where it becomes zero and changes sign below a distance of 1 AU, then increases rapidly in magnitude as R decreases in that range. (bold mine)

They want to propose for a way to test for this. I think it's a good idea.

BTW, I had read somewhere that known physics applied to PA yields an error to 6 orders of magnitude, mentioned at some conference, but I can't find the source. It would be interesting if this were so, still lookin...

[nutant gene 71]

Nutant, I'm pointing out that the equation you say to use for calculating acceleration does not produce the accelerations we observe for any object, anywhere in the solar system. You said to use this: a = (GM a_o/ r^2)^1/2. It does not work:

Acceleration due to the sun's gravity at 1 AU distance:
Newton: a = 5.93E-03 m/s^2
MOND: a = 8.04E-10 m/s^2

At 5 AU distance:
Newton: a = 2.37E-04
MOND: a = 3.60E-10

etc.

This is why MOND doesn't apply to objects experiencing an acceleration greater than a₀.

I see what you mean, and agree that the way you calculated it, the numbers are not right. However, if AUn of (sqrt) 5 is factored in for Jupiter, then the numbers should be closer to what is observed. (Viz., 3.60E-10 times 2.236 = 8.05E-10.) There's a caveat, however, as per what I found on Nieto's report (PDF 1.14 MB):

Even so, there remained one relatively large effect on this scale
that had to be modeled: the solar radiation pressure of the Sun,
which also depends on the craft’s orientation with respect to the
Sun. This effect is approximately 1/30,000 that of the Sun‘s gravity
on the Pioneers and also decreases as the inverse-square of the distance.
It produced an acceleration of ~20 x 10-8 cm/s2 on the
Pioneer craft at the distance of Saturn (9.38 AU from the Sun at
encounter). (For comparison, the gravitational acceleration of the
Sun at the Earth is 0.593 cm/s2). Therefore, any “unmodeled
force” on the craft could not be seen very well below this level at
Jupiter. However, beyond Jupiter it became possible.

The PA does not appear until we are past Jupiter, because of solar wind pressure up until 5 AU, where it begins to drop off.

I guess. If by "mine" you mean that MOND can't produce the Pioneer acceleration, then yes. But it's not like I'm making some startling breakthrough about MOND. MOND doesn't claim to be able to produce the Pioneer acceleration. Which is why I don't understand why you are trying to use it in this way.

No one ever said PA is the same as MOND, matter of fact, it may be 6 orders of magnitude lower than Milgrom's a_0, if it works within our solar system, as shown. This is not a proof it works within our solar system, not until we measure for inertial-gravitational mass beyond 5 AU. So this is only a proposal that it may work this way, where MOND's value for our solar system is down around -a_0s = 1.09E-16 m/s^2 (vs. Milgrom's 1.2E-10 m/s^2).

[Tassel]

So this is only a proposal that it may work this way, where MOND's value for our solar system is down around -a_0s = 1.09E-16 m/s^2 (vs. Milgrom's 1.2E-10 m/s^2).

It was just shown that this doesn't work...

[czeslaw]

I
No one ever said PA is the same as MOND, matter of fact, it may be 6 orders of magnitude lower than Milgrom's a_0, if it works within our solar system, as shown. This is not a proof it works within our solar system, not until we measure for inertial-gravitational mass beyond 5 AU. So this is only a proposal that it may work this way, where MOND's value for our solar system is down around -a_0s = 1.09E-16 m/s^2 (vs. Milgrom's 1.2E-10 m/s^2).

The effect is seen in radio Doppler and ranging data, yielding information on the velocity and distance of the spacecraft. When all known forces acting on the spacecraft are taken into consideration, a very small but unexplained force remains. It causes a constant sunward acceleration of (8.74 ± 1.33) × 10−10 m/s2 for both spacecraft. http://en.wikipedia.org/wiki/Pioneer_anomaly

Pioneer anomaly is abot 8 times stronger then Milgrom's MOND.

[nutant gene 71]

It was just shown that this doesn't work...

Perhaps so, but there are people looking at "Alternative Gravities" at Planetary Society for what this Pioneer Anomaly is about, including how it fits MOND. From their letter:

In fact, many of the presently available alternative gravity models -- in addition to explaining galaxy rotation curves -- predict a small anomalous acceleration of the same order as the one observed with the Pioneers. One such theory is called modified Newtonian dynamics, or MOND, in which the modified behavior kicks in below a certain acceleration -- rather than distance -- scale. Remarkably, the value of this universal acceleration parameter was chosen to allow MOND to describe the dynamics of galaxies, while preserving Newtonian gravity elsewhere. It is intriguing that this small parameter is very close to the anomalous acceleration observed by the Pioneers.

A relativistic extension to MOND, called TeVeS (for Tensor Vector Scalar structure of the model), has all the desirable features of an alternative theory of gravity: it reduces Einstein's theory for high speeds and large accelerations (thereby accounting for gravitational lensing); to Newtonian gravity for low speeds and small accelerations (such as those on Earth); and to MOND when accelerations are smaller still (thereby predicting the observed galaxy-rotation curves).

So others are looking at this, including alternative gravity models that predict the anomalous acceleration towards the Sun by the Pioneers. The fact that they are willing to go over the 30 years of data collected from the Pioneers, with possible future tests for gravity in the outer solar system, shows how important this study may be for a better understanding of gravity and space physics.

[nutant gene 71]

Way back on this post on Jerry’s ATM idea (now closed), I first stumbled upon modifying Milgrom’s into a ‘gentler’ MOND for our solar system, when I wrote it out as, with the modification for G:

Take Milgrom's F = ma^2/ a_o and match it against Newton's other force equation: F = GMm/ r^2, and see what happens.

ma^2/ a_0 = GMm/ r^2

Drop the 'm' on both sides but add a variable for G, where it is 'r' dependent, and bring over 'a_o'. Not orthodox, but bear with me, so here goes:

a^2 = G(r)M a_o/ r^2

This is a way of saying that G is a 'variable' and increasing with distance 'r', so simplifying gives us:

a^2 = GM a_o/ r which then becomes, taking square root:

a = (GM a_0/ r)^1/2

What I was doing was factoring in a ‘hypothetical’ rise in Newton’s G per distance from the Sun ‘r’, but modified it further to reflect it as a function of 1G per 1AU, so that the upper right side became: G(AUn)M a_0, and the lower right side became: r(AUr), which is the equation used in the OP: -a = [G(AUn)M a_0/ r(AUr)]^1/2. The justification was that ‘r’ in the upper is broken down into a function of 1G per 1AU as a number ‘n’, while retaining in the lower part the r^2 function, also modified for 1G per 1AU, where (AUr) is the distance of 1AU, and ‘r’ is the distance from the Sun. This is only a ‘playing around’, but it did yield something useful, that for a ‘modified’ a_0s = 1.09E-16 m/s^2, the Pioneer anomalous acceleration is recreated: -a = ~8E-10 m/s^2.

The purpose for this exercise was to move the above post’s equation of a = (GM a_0/ r)^1/2 into something useful, by retaining the ‘r’ above and below in the final product, which is different from MOND’s equivalent –a = (GM a_0/ r^2)^1/2, by converting ‘r’ into a variable G in the upper as 'n', AUn, and a constant ‘r’, as AUr, in the lower, to match up with the original: a = (GM a_0/ r)^1/2, so that it is once again resembles the real MOND equation of –a = (GM a_0/ r^2)^1/2, though adjusted for 1G per 1AU hypothesis.

However, this leads to another observation in Jerry’s ATM post, which came from the article “Einstein’s Dilemma”:

"I assumed that when the accelerations due to gravitational forces became very small, the formula changes to F = ma²/a0," Milgrom says. According to Milgrom, this change holds only when accelerations fall below one 10-billionth of a meter per second every second. Not only does this modification work best with the data, he adds, but the new constant, a0, may be of cosmological significance: Accelerating at this rate will take you from a resting state to the speed of light in the lifetime of the universe. Otherwise Newton's law operates as usual. So with MOND, stars in the outer reaches of galaxies move faster than expected, not because of the influence of some invisible matter but because Milgrom's amended version of Newton's second law increases the force acting on them.

So here we have an observation by Milgrom, that his “new constant a0, may be of cosmological significance”, since it approximates the Hubble constant, or more specifically, it matches (acceleration to) the speed of light c over the computed age of the universe, 13.7 billion years. What is intriguing about this, and why I bring it up here, is that: if accelerating to velocity ‘c’ over 13.7 billion years matches a0, then what is the relationship between the two telling us? My instinct tells me that there is some sort of correlation between the two, which I suspect matches why light redshifts over 13.7 billion light years into ‘invisibility’ from the visible light spectrum into the infrared. If we look as light redshift over that distance as a function of gravitational effect postulated by MOND for the whole visible universe, then we can postulate that these two values of redshift, the Hubble constant and at what distance it becomes ‘invisible’ should match up as well. In other words, if Newton’s gravity G is some 6 orders of magnitude greater in most of intergalactic space than what we measure as G here, then it should be gravitationally redshifting light at exactly the rate of a0 Milgrom calculated, what should take us back over 13.7 billions years back to its origin. Today, that ‘origin’ is understood as matching the birth of the universe in a Big Bang (where it is ‘expanding’ at the observed Hubble constant), but it need not be if light is gravitationally redshifted naturally, to where at that origin it ceases to ‘exist’ for our observations. However, that said, it leaves open the question of how does light redshift gravitationally over such cosmic distances? If it is ‘climbing out of a gravity well’, how can such a gravity well be understood in any other way than some Machian totality of all the cosmic dust and gases mass, at higher G and equivalence, for all the distance of the known universe through which it traveled? Is that the hypothesized ‘dark matter’ needed to make MOND’s a0, and flat galaxy rotation curves, work? This is too difficult to conceptualize, given our known physics, but it may be a possibility? Really just asking a question out loud, not presenting a new theory of anything, but it is a thought, or more likely a conundrum. So I leave this idea here more as a puzzle that could be resolved, or not, than as a new theory.

Am I making sense here? Or is this like from David Byrnes’ old song “Stop making sense!”…?

[nutant gene 71]

The effect is seen in radio Doppler and ranging data, yielding information on the velocity and distance of the spacecraft. When all known forces acting on the spacecraft are taken into consideration, a very small but unexplained force remains. It causes a constant sunward acceleration of (8.74 ± 1.33) × 10−10 m/s2 for both spacecraft. http://en.wikipedia.org/wiki/Pioneer_anomaly

Pioneer anomaly is abot 8 times stronger then Milgrom's MOND.

You must remember that Milgrom's very small acceleration a_0 is on the right side of the equation, a square root function, while Pioneer Anomaly is on the left side, as -a towards the Sun. So though they resemble each other numerically, they are substantially different. What makes it interesting, however, is that a_0 is closer to Hubble's constant, when figured as acceleration from zero to c over 13.7 billion years, as mentioned in my post above. Cheers.

[czeslaw]

You must remember that Milgrom's very small acceleration a_0 is on the right side of the equation, a square root function, while Pioneer Anomaly is on the left side, as -a towards the Sun. So though they resemble each other numerically, they are substantially different. What makes it interesting, however, is that a_0 is closer to Hubble's constant, when figured as acceleration from zero to c over 13.7 billion years, as mentioned in my post above. Cheers.

It is on right side if you use it as an average acceleration in Universe's expansion.
Milgrom uses it as centripetal acceleration on the left side of the equation, holding the stars in galaxy.

[Nereid]

There will be a new test of Newton's second law published soon, which seems to probe the acceleration regime MOND covers, the Pioneer anomaly, and more no doubt.

The GPB team will also announce their first scientific results this week.

It will be interesting to see how well your ATM idea can account for these two new results (let's hope that this thread isn't closed - 30 day rule - before you've had a chance to check them out!)

[nutant gene 71]

There will be a new test of Newton's second law published soon, which seems to probe the acceleration regime MOND covers, the Pioneer anomaly, and more no doubt.

The GPB team will also announce their first scientific results this week.

It will be interesting to see how well your ATM idea can account for these two new results (let's hope that this thread isn't closed - 30 day rule - before you've had a chance to check them out!)

Cool. Thanks. We still have a couple of weeks, very exciting.

[czeslaw]

There will be a new test of Newton's second law published soon, which seems to probe the acceleration regime MOND covers, the Pioneer anomaly, and more no doubt.

The GPB team will also announce their first scientific results this week.

It will be interesting to see how well your ATM idea can account for these two new results (let's hope that this thread isn't closed - 30 day rule - before you've had a chance to check them out!)

Newton's second law is correct.
The MOND anomalous acceleration is because an additional Dark Matter mass. The Dark Matter effect is because a Vacuum Energy of the gravitational field mass distribution.

[nutant gene 71]

In reference to Garth's post on expected results from GP-B, though off topic, here's my 2 cents worth.

I suspect they will find positive results for the Lense-Thirring effect, to be substantially less than < 0.321E-12 kg m/s. I say 'less than' because in polar orbit the effect from Earth's rotation is felt most at equator and least at poles. (I don't know how to translate this into arcsec/yr, so leave it at that.) So expect positive Frame-dragging effect, but not sure any meaningful results from Geodetic effect, so put that 0.0000 arcsec/yr. Well, that's my call, for little it's worth.

Of course, this does not explain in any way the possible MOND effect for our solar system, but it's pretty cool in its own right. We will see hopefully the finally tally soon.

[nutant gene 71]

Well blow me down. The Geodetic effect showed a positive non-zero effect. So scratch my above, retracted, obviously worthless statement.

There’s still the more important very small Frame-dragging effect to be released, per http://einstein.stanford.edu/index.html . Will the Lense-Thirring effect show up as expected? That’s still ‘up in the air’ and a big one, so awaiting the small numbers to be released December.

Or as said by Jerry on another post:

The problem here is, the G-Probe B scientists cannot bring the shell down and confirm that there are magnetic 'patches' in the supporting framework that are causing unexpected deviations. Likewise, they cannot confirm there is a viscous element in each of the gyroscopes that they failed to observe in two decades of ground-based testing. … Reading between the lines, it was during this final calibration that the evidence of the 'patch effect' emerge, so they do have a pretty good method for nulling this unexpected source of error. Great experiment! Great, puzzling results!

We shall see come December. So far, score another one for Einstein! The universal constant G is safe for now.