Mercury's precession is because of GR, but if no GR what would planet Vulcan's orbit roughly be?

[repoman]

I have known about why planet Vulcan was hypothesized in the 1840s for a while now, and a lot of the story behind the pointless searches for it and finally how General Relativity "killed" it. It is one of the best stories in all of astronomy.

http://www.pbs.org/wgbh/nova/next/physics/hunt-for-vulcan/

But if Newtonian physics explained everything where would Vulcan be and what size roughly?

If lpetrich sees this I am pretty sure he would be able to give a close approximation.

 

[beero1000]

You won't be able to work out a single solution, as there are many possibilities (that's why there was such trouble trying to find it), but I'll give it a shot...

The observed precession for Mercury's orbit is 574 arc seconds per century, and about 531 of those are accounted for by Newtonian effects, so we need to explain 43 arc seconds per century's worth of perihelion precession. That means we need to explain about 0.05 arc seconds in apsidal angle.

As a first order approximation (and assuming planets are circular rings of constant linear density ), I get that a planet Vulcan of mass \(M\) with an orbital radius ratio \(R\) (as a proportion of Mercury's orbital radius, \(\frac{R_m}{R_v}\)) would have that effect if:

\(\frac{M R^3}{R^2 - 1} \approx 4.812 \cdot 10^{23} \text{ kg}\)

So a planet orbiting the sun at half the distance as Mercury would need to have a mass of about \(1.8 \cdot 10^{23} \text{ kg}\), or a little more than half of Mercury's mass. The closer Vulcan would orbit to Mercury, the less mass it would need to have the desired effect, which makes sense.

 

[repoman]

cool thanks.

The funny thing about this is that there have to be some conspiracy theorists who are proficient at Newtonian dynamics and deny GR that have run these numbers as well.

You would think they would say that Vulcan might be in a 1:2 or 2:3 orbital resonance with mercury - even though that would show up really obviously with Mercury speeding up and slowing down in a resonant cycle. In fact, what did Leverrier have to say about the lack of this for Mercury's orbit?

Maybe I am wrong, this is all modeling game anyway.

 

[lpetrich]

Here, "inner" and "outer" are relative to a planet's orbit, so Venus is an outer planet relative to Mercury and Jupiter an inner planet relative to Saturn.

From celestial mechanics, the lowest-order precession rates are

w(prec) = (3/4)*(m/msun)*w(orb)*(ax/a)² for an inner planet and

w(prec) = (3/4)*(m/msun)*w(orb)*(a/ax)³ for an outer planet.

(pericenter direction: forward, nodes: backward) (planet's major axis: a, other planet's major axis and mass: ax, m)

The general expression is this rather nightmarish integral:

w(prec) = (1/2)*(m/msun)*w(orb)*a²*ax * (1/pi)*integral of cos(p)/(a² + ax² - 2*a*ax*cos(p))^3/2 over p from 1 to pi

That can be done with elliptic integrals, though it is easy to calculate it numerically.

For completeness, the effects of the other planet's eccentricity and inclination must also be included, but I'm ignoring that for now. They have similar expressions, however.

-

Now for Mercury's excess perihelion precession. It is 43 seconds of arc per century, or 8.0*10^-8 of Mercury's average angular velocity or "mean motion".

That is in very good agreement with general relativity, which predicts (3*gravconst*msun)/(a*(1-e²)*c²) = 8.0*10^-8 here.

But here are estimated planet masses as a function of relative major axis:

• 0.75 -- 0.013
• 0.5 -- 0.082
• 0.25 -- 0.50
• 0.2 -- 0.82
• 0.15 -- 1.5
• 0.1 -- 3.5

So an intra-Mercurian planet that makes that extra precession should be large enough to observe.

But if it is a lot of asteroids instead, those asteroids will be more difficult. But from recent observations of the Sun, their upper limit of size must be about 20 km.

 

[repoman]

Thanks, lpetrich.

what are the units of the planet masses and the major axis?


For having there be an inferior orbit planet to Mercury (call it Vulcan) that would be an ok fit for the excess perihelion, wouldn't there being slow down of Mercury's velocity as Vulcan caught up with Mercury and a speed up after it passed?

This is a diagram showing what I mean





I wonder if there was then (mid 1800s) enough precision to detect this. Were people like Leverrier looking for this?

 

[lpetrich]

Thanks, lpetrich.

what are the units of the planet masses and the major axis?

The Earth's mass and Mercury's major axis.

For having there be an inferior orbit planet to Mercury (call it Vulcan) that would be an ok fit for the excess perihelion, wouldn't there being slow down of Mercury's velocity as Vulcan caught up with Mercury and a speed up after it passed?

...

I wonder if there was then (mid 1800s) enough precision to detect this. Were people like Leverrier looking for this?

Such short-period effects do exist, and that is how Urbain Leverrier and John Couch Adams predicted where Neptune would be. But for Mercury, it is much more difficult. The effect size is about 8*10^-8 radians or 0.016 seconds of arc -- too small to be observed in the 19th cy. and difficult to observe nowadays.