Mercury Precession & GR Calculation Impact

[D H]

That is a strawman, starthaus. The authors said

If $ds_1$ and $ds_2$ are the arc lengths subtended by differential angular element $d\alpha$ and if $a$ is small compared to R¹¹

$$dm_i = \lambda ds_i \simeq l_i \lambda d\alpha \qquad\qquad(2)$$

The astute reader will note that Eq. (2) will not be valid for values of $a\sim R$, since the angle $d\alpha$ would subtend a mass larger than $ds_i$ by a factor of $(\cos \alpha)^{-1}$. Since the orbits of Venus and Mercury differ by approximately a factor of 2, we repeated the calculation including the $(\cos \alpha)^{-1}$ term. An exact solution could not be found, but a series expansion to terms of order $(a/R)$ showed that the errors introduced by ignoring the $(\cos \alpha)^{-1}$ term were of order of 2.3%.​

In other words, the authors explicitly acknowledged that this is an approximation and they checked the validity of the approximation.

 

[starthaus]

That is a strawman, starthaus. The authors said

If $ds_1$ and $ds_2$ are the arc lengths subtended by differential angular element $d\alpha$ and if $a$ is small compared to $R$¹¹

$$dm_i = \lambda ds_i \simeq l_i \lambda d\alpha & (2)$$

The astute reader will note that Eq. (2) will not be valid for values of $a\sim R$, since the angle $d\alpha$ would subtend a mass larger than $ds_i$ by a factor of $(\cos \alpha)^{-1}$. Since the orbits of Venus and Mercury differ by approximately a factor of 2, we repeated the calculation including the $(\cos \alpha)^{-1}$ term. An exact solution could not be found, but a series expansion to terms of order [itex](a/R)[/itex[ showed that the errors introduced by ignoring the $(\cos \alpha)^{-1}$ term were of order of 2.3%.​

In other words, the authors explicitly acknowledged that this is an approximation and they checked the validity of the approximation.

Yes, I pointed that out. They acknowledged that it is a hack that isn't valid for a~R. It is a hack that it is never valid since it defies elementary geometry (the kind they teach in 9-th grade). You know where you can find a decent treatment.

 

[George Jones]

The author's intent in this paper is "to show that the component of the precession of Mercury's perihelion due to the outer planets can be calculated easily at the undergraduate level." He did so quite admirably IMO. The mathematics is quite simple, and yet comes remarkably close to those obtained by "more advanced treatments."

Yes, I agree.

The authors have inadvertently canceled out the square roods

You have not acknowledged that you made a mistake here.

Yes, I pointed that out. They acknowledged that it is a hack that isn't valid for a~R. It is a hack that it is never valid since it defies elementary geometry (the kind they teach in 9-th grade). You know where you can find a decent treatment.

I don't like your tone. The authors have not made a mistake, they have made a useful approximation. This is the kind of skill to which students should be exposed, because this type of thing is ubiquitous in physics. If an approximation were equal to the unapproximated expression, it wouldn't be an approximation.

It is a useful exercise to express the difference between

http://www.physics.princeton.edu/~mcdonald/examples/mechanics/price_ajp_47_531_79.pdf

and

http://farside.ph.utexas.edu/teaching/336k/Newton/node127.html

in terms of powers of r/a.

 

[starthaus]

It is a useful exercise to express the difference between

http://www.physics.princeton.edu/~mcdonald/examples/mechanics/price_ajp_47_531_79.pdf

and

http://farside.ph.utexas.edu/teaching/336k/Newton/node127.html

in terms of powers of r/a.

Sure, it is. Or you can use the algorithm that I outlined in calculating the exact integral describing the resultant force exerted by a ring. Your choice.