Does the expansion of the Universe affect orbiting bodies?

[PeterDonis]

How would you describe the solution when it comes to the local group of galaxies and, in particular, the collision of the Milky Way and Andromeda?

The local group is a bound system surrounded by empty space (unless we include dark energy, but on this scale dark energy effects are still too small to matter). So it would be modeled as a bound system surrounded by empty space.

The distances here are significant if we plug in the current universe expansion rate.

You can multiply any distance you like by the Hubble constant and get a speed. That in itself doesn't mean the speed has any physical significance.

The correct criterion, as I've already said, is what matter is actually present in the region of spacetime you are talking about. The Local Group of galaxies, as above, is a bound system surrounded by empty space. There is no "expansion velocity" in such a system. The fact that the FRW model for the universe as a whole has a Hubble constant that I can multiply by, say, a million light-years to get a speed, is irrelevant, since the FRW model is simply not a correct model for a bound system surrounded by empty space.

That expansion rate is based on the overall universal balance of mass/vacuum.

More precisely, it's based on assuming that the universe is filled with a uniform density perfect fluid, with the density given by the average density for the universe as a whole.

How would the universe expansion rate get into the equations for the dynamics of the local group?

It wouldn't, since the local group is not composed of a uniform density perfect fluid.

should we model the local group using the specific mass/vacuum density we find in this region?

Using the specific mass/vacuum distribution we find in the region, on whatever length scales are relevant for the question we are trying to answer.

Can these two models be tested against the observed separation velocity?

The FRW model is obviously false for the Milky Way and the Andromeda galaxy, since they are moving towards each other whereas the FRW model says they should be moving apart.

 

[PeroK]

The FRW model is obviously false for the Milky Way and the Andromeda galaxy, since they are moving towards each other whereas the FRW model says they should be moving apart.

What's being proposed, as I understand it, is a hybrid model where we simply layer a universal expansion rate on top of the local dynamics. This would reduce very significantly but not eliminate the attraction between the local galaxies. This seems to be a widespread misapprehension.

 

[Ibix]

What's being proposed, as I understand it, is a hybrid model where we simply layer a universal expansion rate on top of the local dynamics. This would reduce very significantly but not eliminate the attraction between the local galaxies. This seems to be a widespread misapprehension.

I agree with your interpretation of what's being proposed, and it's not an unreasonable reading of "space is expanding". Unfortunately, it's wrong. It's certainly possible that dark energy (or a similar phenomenon) is pushing outwards slightly on the solar system and/or Andromeda and the Milky Way, but that would be below our detection level. And dark energy (etc) isn't responsible for the expansion of the universe - it's a tiny tweak to the expansion profile. So while it might be correct to say that dark energy has an effect on the motions of planets/stars/galaxies, that's not the same as saying expansion has an effect on their motions.

 

[PeterDonis]

What's being proposed, as I understand it, is a hybrid model where we simply layer a universal expansion rate on top of the local dynamics.

Proposed by whom? What are you referring to?

 

[PeroK]

Proposed by whom? What are you referring to?

To see just how significant the effect is, the distance to Andromeda is two and a half million light years. The current value of the expansion rate is 70km/sec per megaparsec, so the expansion contributes 55km/sec to Andromeda's recession from the Milky Way. But these two galaxies are supposed to collide in five billion years, approaching each other with a velocity of 110km/sec. What this means is that, were the universe not expanding, the two galaxies would be approaching each other at a speed of 165km/sec and collide in three billion years instead. I have no idea whether or not this expansion has been taken into account in the calculation of the collision's timescale.

 

[PeterDonis]

Proposed by whom?

Ah, ok. I think I've already explained in quite a bit of detail what is wrong with this proposal.

 

[PeterDonis]

it's not an unreasonable reading of "space is expanding"

Not unreasonable if all you have looked at is pop science sources, perhaps. But that just underscores the point that you should not be trying to learn science from pop science sources.

 

[Buzz Bloom]

ERRORS HAVE BEEN FIXED.

If the universe were everywhere filled with a perfect fluid (this is made up of genuinely continuous matter, not of atoms or anything - don't ask too many questions) that was everywhere of uniform density and pressure, then that universe would be accurately described by an FLRW spacetime.
...
The real universe is rather like this at very large scales.
...
I think your attempt to understand "bound systems" in an expanding universe is probably the wrong way to look at this.

Yes, you're mistaken. "Bound" is a state of the system of bodies, not of one body or the other. The escape velocity criterion is the escape velocity from the system, based on the total mass of the system.
...
In other words, what "escape" means, and therefore what "bound" means, depends on what system you are looking at.
...
The McVittie metric itself is highly unrealistic for analyzing bound systems for the same reason FLRW metrics in general are: it assumes that there is a uniform density of matter everywhere, except for (in the McVittie case) a single "mass" at the spatial origin. But that is not true of the actual universe.

Hi Ibiz and Peter:

I do appreciate your posts, but I still feel the need to ask a few followup questions.

Am I wrong to assume that the math of various models are able to make reasonable predictions about the behavior of the models, even if the models have flaws with respect to the real universe? I certainty understand that the McVitty model, combining Schwartzschild and FLRW math, is not a realistic model of our real universe. Suppose I take the simplest case in which there is a single black hole mass, and a FLRW model as follows:
MODEL 1:

M = the mass of our sun.​

Ω_R is very small, say the value in our actual current universe, say​

Ω_R = ε,​

Ω_R is very small, say the value in our actual current universe, say​

Ω_M = 0,​

Ω_k = 0,​

and

Ω_Λ = 1-ε.​

Assume a spherical ball of spherically symmetric mass equal to mass m of Earth.
Assume mass m is in a circular orbit around mass M as a distance R of exactly

R = 149.60×10⁶ km.​

As I understand properly what I read in Wikipedia, this circular orbit would have an average orbital velocity very close to

V = 29.78 km/s.​

The reason I have a non-zero Ω_R is so that there can be a CBR which is useful in determining whether a particle is stationary with respect to co-moving coordinates, or if moving, the velocity can be measured based on differences in CBR differences in different directions
MODEL 2:
(This model previously had several serious errors which are now fixed.)
The model is the same as MODEL 1 except for the following differences.

Ω_k = η<0, 0<ε<|η|<<1.​

Ω_M = 1-ε-η>1​

assumed to be uniformly distributed dark matter with no forces acting between particles,
and

Ω_Λ = 0.​

For this model the it is possible for the universe to (at least for a while) stay at a constant scale factor, which means no expansion would be taking place.

What I would like to find out, either from someone else, or if necessary from my own efforts to work with the McVitte metric, is whether or not the MODEL 2 orbit would be identical with the MODEL 1 orbit. My current uniformed guess (which certainly may be wrong) is that MODEL 2 would have a very small but non-zero difference from MODEL 1. In particular the MODEL 2 orbital radius would be slightly smaller, and the orbital velocity slightly faster.

I also want to explore MODEL 1 in which the distance D between the M and m masses satisfies

D³ = GM/H².​

BTW, a very rough calculation is that this D value is about 3000 light years. I am wondering is the McVitty math can show that at the exact distance D the MODEL 2 orbital velocity would be zero.

Regards,
Buzz

 

[PeterDonis]

Am I wrong to assume that the math of various models are able to make reasonable predictions about the behavior of the models

Of course not. You can always get mathematical predictions from the models. But if a model is already known to not be a good model for the actual universe, I don't see the point of spending a lot of time trying to calculate predictions from it. You'll just be calculating a lot of things that don't match up to our actual universe.

 

[PeterDonis]

For this model the universe it is possible for the universe to (at least for a while) stay at a constant scale factor

I don't understand; your MODEL 2 appears to be a matter-dominated universe with no dark energy. It is impossible for such a universe to stay at a constant scale factor "for a while". At most, if it is closed, it will be at a constant scale factor for an instant, at maximum expansion.

 

[PeterDonis]

MODEL 1

This appears to be a dark energy dominated universe with no matter and a smidgen of radiation. (Btw, the value of $\epsilon$ that would approximate our current CMBR as compared to the rest of the matter and dark energy in our actual universe is about $10^{-4}$ IIRC.) In such a universe, the presence of dark energy does indeed have a tiny effect on the orbital parameters of an object like the Earth orbiting the Sun. The simplest way to view the effect is that it makes the orbital velocity required to maintain orbit at a given altitude very slightly smaller than it would be in the absence of dark energy. The presence of the radiation has an even smaller effect in the opposite direction, i.e., requiring a very, very slightly larger orbital velocity at a given altitude.

 

[Buzz Bloom]

Hi @PeterDonis :

Thank you for your responses. I have realized that I made (at least) two errors in my post #43. I am going to (try to) fix them before I respond to your recent posts. I apologize for my carelessness.

Regards,
Buzz

 

[Buzz Bloom]

Of course not. You can always get mathematical predictions from the models. But if a model is already known to not be a good model for the actual universe, I don't see the point of spending a lot of time trying to calculate predictions from it.

I interpret the above as being a confident statement that a conclusion reached by these unrealistic models, cannot possibly be useful for a realistic model. I suggest a hypothetical counter view. In particular, suppose hypothetically the exploration of a non-relaistic model like my MODEL 1 based on the McVitte metric demonstrates that for D satisfying

D³ = GM/H²​

that Earth mass m is metastable (stable unless a perturbation destroys the stability) at zero velocity relative the mass M. Would not that then suggest that something (perhaps an effect of expansion) is apposing the gravitation acceleration on m due to M? Would this then also suggest that there might be a corresponding effect in a more realistic model?

Sorry, Peter. I have to stop posting for a while. My wife needs to use this computer now for several hours.

Regards,
Buzz

 

[PeterDonis]

I interpret the above as being a confident statement that a conclusion reached by these unrealistic models, cannot possibly be useful for a realistic model.

Wrong conclusions are not useful, yes.

Would not that then suggest that something (perhaps an effect of expansion) is apposing the gravitation acceleration on m due to M?

I've already said that, since your MODEL 1 is dark energy dominated, it does predict a tiny force tending to push things apart in the solar system. This is an obvious prediction of any dark energy dominated model. It is also irrelevant in practice to the dynamics of the solar system, for reasons already given (the effect is much too small in practice to be detectable).

So your MODEL 1 is not telling anyone anything they don't already know or making any practically useful prediction. But the reason has nothing whatever to do with the model being "unrealistic"; as I've already said, in the case of dark energy, we do think its density is uniform everywhere, so your MODEL 1 is perfectly realistic in that respect. It's just that the density of dark energy is so tiny that its effects are unmeasurable on solar system scales.

The model that I have been saying is "unrealistic" is a model like the McVittie metric for the matter dominated case, where the model assumes that there is a uniform density of ordinary matter everywhere in the universe, and tries to make claims about the dynamics of the solar system on that basis. I do not think the predicted "expansion" effects in such models exist at all on the scale of the solar system, precisely because we already know that there isn't a uniform density of ordinary matter on such scales; there are isolated stars and planets surrounded by empty space. And we already know how to model that using GR, and such models do not have any "expansion" effects.

 

[Buzz Bloom]

I've already said that, since your MODEL 1 is dark energy dominated, it does predict a tiny force tending to push things apart in the solar system. This is an obvious prediction of any dark energy dominated model. It is also irrelevant in practice to the dynamics of the solar system, for reasons already given (the effect is much too small in practice to be detectable).

Hi Peter:

I interpret the above as saying that you agree that in the McVittie Model 1 there is a distance D_lim such that

(Eq 1) D_lim³ approx = GM/H²​

at which the orbital velocity of the Earth object relative to the Sun object is zero. If this is correct this means that the definition of "boundeness" based on the escape velocity V defined as

(Eq 2) V² = 2GM/D​

for any distance D is incomplete. A more complete test would include that if a particle is at a distance greater than the D_lim of (Eq 1) is is not bound to the central mass M, no matter what it's velocity is. Also, for any separation distance

D < D_lim,​

the escape velocity V (Eq 2) is too large. It needs to be reduced by an amount which can be calulated based on what you call the effect of Ω_Λ.

Regards,
Buzz

 

[Buzz Bloom]

I don't understand; your MODEL 2 appears to be a matter-dominated universe with no dark energy. It is impossible for such a universe to stay at a constant scale factor "for a while". At most, if it is closed, it will be at a constant scale factor for an instant, at maximum expansion.

Hi Peter:

You are correct, and I am wrong, and I apologize for my mistake. I recall working on calculating the range of Omega values that would allow for

(1) da/dt=0​

and

(2) d²a/dt² = 0.​

If I remember correctly I did this about two years ago, and I had forgotten the need for dark energy. Aside from a diminishing of math skills, my memory is also not as good as when I was younger.

However, I believe this model is sufficient for my purposes with a value of a satisfying (1). I want to calculate what the difference is between the Model 1 and Model 2 orbits.

Regards,
Buzz

 

[PeterDonis]

I interpret the above as saying that you agree that in the McVittie Model 1 there is a distance Dlim such that
(Eq 1) Dlim3 approx = GM/H2at which the orbital velocity of the Earth object relative to the Sun object is zero.

Meaning, in the Schwarzschild-de Sitter metric, since that is what "the McVittie Model 1" is--zero matter, zero radiation, only a cosmological constant. Yes, the statement is true, although I would phrase it somewhat differently, as I did in some thread or other recently (we have several threads now on the same general topic), that there will be some distance from the central mass at which an object can "hover" at a constant altitude with zero proper acceleration, i.e., by simply floating freely in space.

if a particle is at a distance greater than the Dlim of (Eq 1) is is not bound to the central mass M, no matter what it's velocity is

No, this is not correct. The fact that the object can "float" at a constant altitude does not mean it can escape. You would still have to add energy to it for it to escape, where "escape" means "move away from the central mass indefinitely, without ever stopping or falling back".

But there is a more important issue that you are ignoring: in our actual universe, there are multiple isolated gravitating systems in a hierarchical structure, so the idea of "escape" has to be put in context. What does it mean to "escape" from the solar system? If I am halfway between the Sun and Alpha Centauri (supposing for a moment that their masses are equal), have I "escaped"? And even if I am well away from all stars in the immediate neighborhood, I'm still bound inside the Milky Way Galaxy. And even if I "escape" from the galaxy, I'm still bound inside the Local Group. And even if I "escape" from that, I'm still bound inside a larger galaxy cluster.

You are ignoring all of this hierarchical structure, and that's not really valid.

 

[PeterDonis]

for any separation distance
D < Dlim,the escape velocity V (Eq 2) is too large.

I gave a formula for the case where the effects of dark energy are included in some post or other that's been linked to. This discussion is too spread out between multiple threads for me to find it right now.

 

[PeterDonis]

I want to calculate what the difference is between the Model 1 and Model 2 orbits.

First you have to decide how you are going to find "the same" orbit in each model, in order to compare them. How will you do that?

 

[Buzz Bloom]

First you have to decide how you are going to find "the same" orbit in each model, in order to compare them. How will you do that?

Hi Peter:

My thought about this is that the Sun mass M and Earth mass m are the same in the two MODELs. If I calculate the circular orbit velocity for each of the two MODELs, assuming the same distance D between the Earth and Sun, I expect the two orbital velocities to be slightly but non-zero different.

However, I still need to learn how to calculate a circular orbit given the metric. Any suggestions of what I should read would be appreciated.

Regards,
Buzz

 

[PeterDonis]

My thought about this is that the Sun mass M and Earth mass m are the same in the two MODELs. If I calculate the circular orbit velocity for each of the two MODELs, assuming the same distance D between the Earth and Sun, I expect the two orbital velocities to be slightly but non-zero different.

Ok, so to be sure I understand, the models are:

MODEL 1: Sun mass M, nothing but dark energy everywhere else, Earth in circular orbit at proper distance D from the center of the Sun.

MODEL 2: Sun mass M, nothing but ordinary matter (i.e., uniform energy density, zero pressure) everywhere else, Earth in circular orbit at proper distance D from the center of the Sun.

And just for comparison:

MODEL 3: Sun mass M, vacuum everywhere else, Earth in circular orbit at proper distance D from the center of the Sun.

Is the above correct?

 

[PeterDonis]

Is the above correct?

Note, btw, that I carefully said "proper distance D" instead of "radial coordinate D", since they're not the same.

 

[PeterDonis]

MODEL 3: Sun mass M, vacuum everywhere else, Earth in circular orbit at proper distance D from the center of the Sun.

I should also note, btw, that as far as the Earth's orbit about the Sun is concerned, MODEL 3 is exactly equivalent in all its predictions to:

MODEL 3A: Sun mass M, vacuum out to some proper distance A that is larger than the proper distance D of Earth from Sun in its circular orbit, and any spherically symmetric distribution of stress-energy outside proper distance A. (And if we substitute "solar system", and distance A is large enough to be larger than the distance of any object in the solar system from the Sun, the same applies, and similarly for any bound system, up to and including galaxy clusters.)

 

[Buzz Bloom]

Ok, so to be sure I understand, the models are:
...
Is the above correct?
...
Note, btw, that I carefully said "proper distance D" instead of "radial coordinate D", since they're not the same.

Hi Peter:

Those three models, and perhaps a few more I haven't yet decided on yet, are what I currently have in mind to calculate. So I agree that the models you listed "are correct".

I did notice your use of "proper distance", and I agree that may be correct distance I should use. Thank you for mentioning that. However, I am not sure the proper distance from the center of the sun is able to be calculated. It may be necessary to use instead the proper distance from the Sun's Schwartzschild radius (usually denoted as r_s) to the Earth center, or maybe to the Earth's r_s. Or maybe using the coordinate r distance would be OK. After I learn about calculating a circular orbit's velocity, I expect I will be able to chose a reasonable definition for "distance".

Regards,
Buzz

 

[PeterDonis]

However, I am not sure the proper distance from the center of the sun is able to be calculated.

Sure it can. It just can't be expressed in a closed-form formula, unless you make the highly unrealistic assumption that the Sun's density is constant everywhere inside it. But reasonable approximations can be made, especially for the weak field case, which should be sufficient for this discussion.

It may be necessary to use instead the proper distance from the Sun's Schwartzschild radius

The Sun isn't a black hole, so this radius is irrelevant since it doesn't pick out any location of physical significance (same for the Earth).

 

[Buzz Bloom]

The Sun isn't a black hole, so this radius is irrelevant since it doesn't pick out any location of physical significance (same for the Earth).

Hi Peter:

Apparently you missed an assumption regarding the MODELs from my post #34.

Suppose I take the simplest case in which there is a single black hole mass, and a FLRW model as follows:

Regards,
Buzz

 

[PeterDonis]

Apparently you missed an assumption regarding the MODELs from my post #34.

Ah, ok. If the central mass is indeed a black hole, then yes, "proper distance to the center" is not well-defined so you'll need to come up with some other criterion for "same orbit". The Schwarzschild radial coordinate $r$ would be an obvious simple one to use (i.e., "same orbit" means "same $r$"), but $r$ does not directly represent proper distance in the radial direction (for example, the proper distance to the horizon is not the difference in radial coordinates, $r - 2M$).

 

[Buzz Bloom]

the proper distance to the horizon is not the difference in radial coordinates, r−2M

Hi Peter:

I think I understand what M is supposed to represent, but I am not sure. It seems logical that M represents r_s, with some unit constants omitted like G and c. (I feel that I am just not comfortable with equations with missing unit constants.) Is this correct?

I have been working on integrating ds in the Schwartzschild metric from r_s to r assuming that ds and dr are the only non-zero components. I want to compare s with r, for example w/r/t a circular approximation of Earth's orbit. I am making some progress, but I think I have made some math errors, so I am not quite ready to post my results.

Regards,
Buzz

 

[PeterDonis]

It seems logical that M represents r_s,

Actually half of $r_s$. In conventional units what I was calling $M$ would be $GM / c^2$, and the Schwarzschild radius corresponding to $M$ in conventional units is twice that.

I have been working on integrating ds in the Schwartzschild metric from rs to r assuming that ds and dr are the only non-zero components.

Yes, this will give you the proper distance from the horizon to an object "hovering" at $r$. You can then compare it with $r$ itself to see how much error there is in just treating $r$ as "radial distance". For the case of the Earth in orbit about the Sun (or a black hole in place of the Sun), the error will be tiny.

 

[Buzz Bloom]

Ah, ok. If the central mass is indeed a black hole, then yes, "proper distance to the center" is not well-defined so you'll need to come up with some other criterion for "same orbit". The Schwarzschild radial coordinate $r$ would be an obvious simple one to use (i.e., "same orbit" means "same $r$"), but $r$ does not directly represent proper distance in the radial direction (for example, the proper distance to the horizon is not the difference in radial coordinates, $r - 2M$).

Hi Peter:

I have been working on integrating a simplified Schwartzschild metric in which only ds and dr are not zero. This leads to

$ds = \frac {dr} {\sqrt {1-{r_s}/r}}$

When r = r_s, ds = ∞.

This ∞ remains in the value of the integral of ds integrating from r_s to some larger value of r. Therefore it appears to be impossible to integrate ds from r_s.

Is the correct? If so, there is no obvious choice but to use the coordinate r as the variable represening the distance from a gravitating body to a circular orbit. Do you agree? If not, how about using 3r_s/2 as the lower choice for the integration. Wikipedia says this is the orbit with orbital velocity c.

When calculating the proper distance between circular orbits in the same plane, it would be possible to integrate ds.

Regards,
Buzz

 

[PeterDonis]

it appears to be impossible to integrate ds from r_s.

No, it isn't. Look up the integrand in a table of integrals; you will see that it has a well-defined antiderivative even for $r = r_s$, so the antiderivative can just be evaluated at both endpoints and the difference taken to obtain the answer. This is a good example of why you shouldn't use your intuition in calculus, at least not if your intuition is untrained.

 

[nrqed]

It's not a matter of "what length scale". It's a matter of whether a particular object is part of a gravitationally bound system or not.

So if two masses, let's say both of 1 kg, are placed at rest relative to one another say 1 Mpc away from one another and there are no other gravitational forces acting on them. The expansion of the Universe will have no effect on their separation because they are gravitationally bound? Their subsequent motion will be the same as if they were in a static Universe? The proper distance between them will vary with time the same way in both cases?

And if say we give a tiny amount of kinetic energy to one of the masses so that they are no longer gravitationally bound, then in that case the expansion of the universe will affect their separation?

 

[PeterDonis]

if two masses, let's say both of 1 kg, are placed at rest relative to one another say 1 Mpc away from one another and there are no other gravitational forces acting on them.

"No other gravitational forces acting on them" is highly unlikely in the actual universe if they are 1 Mpc apart, because it requires that there are no other gravitating objects between them. (It also requires that the distribution of matter outside of them is spherically symmetric. And that there is no dark energy present.) But for an idealized thought experiment, ok.

The expansion of the Universe will have no effect on their separation because they are gravitationally bound?

No, the expansion of the Universe will have no effect on them because there are no other gravitational forces acting on them. You have eliminated all possible effects of any other matter with that specification. That includes the matter in the rest of the expanding universe.

Their subsequent motion will be the same as if they were in a static Universe?

A static universe containing any matter at all (or radiation or dark energy) is impossible, except for the edge case of the Einstein static universe, which is unstable against small perturbations (like a pencil balanced on its point). But if by "static universe" you mean "a universe entirely empty except for the two objects", then yes, your statement is correct.

if say we give a tiny amount of kinetic energy to one of the masses so that they are no longer gravitationally bound, then in that case the expansion of the universe will affect their separation?

Not as long as the "no other gravitational forces" condition holds.

Say, for example, that the two objects are 1 Mpc apart and are inside a "void" containing no other gravitating objects, and the void is 2 Mpc wide. Then if you give the objects just enough kinetic energy so they move apart at their relative escape velocity (so they're no longer gravitationally bound), their motion will be the same as if they were in an empty universe (as I defined that term above) until they are no longer inside the void and there is other matter exerting gravitational force on them. (There is a technicality here: if the other matter outside the void is expanding with the rest of the universe, the void will be wider than 2 Mpc by the time the objects reach its edge. So they will have to be somewhat further than 2 MPc apart to be no longer inside the void.)

 

[Buzz Bloom]

No, it isn't. Look up the integrand in a table of integrals; you will see that it has a well-defined antiderivative even for , so the antiderivative can just be evaluated at both endpoints and the difference taken to obtain the answer.

Hi Peter:

I have gone over my integration work again, and I found my mistake.

I did lookup integrals - in the 11th edition of CRC Mathematical Tables (1957) section on Integrals (pp 274-307). In particular I used the integral forms #60 and #57 on page 279. In what follows, I substitute r for x.
The group of equations is called:

"FORMS CONTAINING $\sqrt {a+br} = \sqrt {u}$ AND $\sqrt {a'+b'r} = \sqrt {v}$ WITH k=ab'-a'b".​

So with

a = -r_s​

b = 1​

a' = 0​

b'= 1​

k = 1​

the metric becomes

$ds = {\frac {\sqrt v} {\sqrt u} } dr$​

Form #60 is

$\int {\frac {\sqrt v} {\sqrt u} } dr = {\frac {1} {b}} \sqrt {uv} - {\frac k {2b}} {\int {\frac 1 { \sqrt {uv}}} }dr$.​

The integral on the RHS uses form #57, which is

${\int {\frac 1 { \sqrt {uv}}} }dr = \frac 2 {\sqrt {bb'}} {log ({\sqrt {bb'u} + b{\sqrt v})}}$​

Combining these two forms produces
$s = {\frac {1} {b}} \sqrt {uv} - {\frac k {2b}} {\int {\frac 1 { \sqrt {uv}}} }$
$s = {\frac {1} {b}} \sqrt {uv} - {\frac k {2b}} { \frac 2 {\sqrt {bb'}} {log ({\sqrt {bb'u} + b{\sqrt v})}}}$
$s = \sqrt {uv} - log(\sqrt u + \sqrt v)$
$s = \sqrt {r(r-r_s)} - log(\sqrt {(r-r_s)} + \sqrt r)$

Now, if the lower limit of the integral is r=r_s, then
$s(r) = \sqrt {r(r-r_s)} - log(\sqrt {(r-r_s)} + \sqrt r) + log(\sqrt {r_s})$

Thank you for telling me about my error.

Regards,
Buzz

 

[PeterDonis]

You have a sign error and a missing coefficient in the log terms. My go-to online reference for integrals is here:

http://integral-table.com/downloads/single-page-integral-table.pdf

Equation (25) is the relevant one, with $x = r$ and $a = - r_s$. The integral then becomes

$$
s(r) = \sqrt{r \left( r - r_s \right)} + r_s \left( \ln \left[ \sqrt{r} + \sqrt{r - r_s} \right] - \ln \left[ \sqrt{r_s} \right] \right)
$$

Which can be rewritten as

$$
s(r) = r \sqrt{1 - \frac{r_s}{r}} + r_s \left( \ln \left[ \sqrt{\frac{r}{r_s}} + \sqrt{\frac{r}{r_s} - 1} \right] \right)
$$

which makes it clearer what happens for $r_s << r$.