How do we interpret measurements of Mercury's position?

[djanni_unchained]

When scientists measured the position of Mercury in the 18th century, they interpreted the results assuming a Euclidean background, because they did not know general relativity. So they measured r and φ in fuction of time attributing to these coordinates an Euclidean meaning, that is, assuming that the relations of Euclidean geometry hold for these coordinates.

In Schwarzschild solution (ds^2=(1−2m/r)dt^2−1/(1−2m/r)dr^2−r^2dΩ^2) r and φ dosn't have the same meaning of the Euclideian r and φ, in mind of scientists of the 18th century; so my doubt is:

if general reltivity predicts a precession of Mercury perielium with Δφ=43'', why we should compere this quantity whit the Δφ measured in 18th century, surely measured with reasoning that assumes a Euclidean space?

 

[Ibix]

##\phi## does have the same meaning, actually, so that isn't an issue. And we're looking at a rate of change of a ##\phi## coordinate with time, so ##r## isn't relevant (it appears in the derivation of the error, but we don't need to measure ##r##, just the ##\phi## where it's a minimum). It's only the ##t## that matters - are you using Newton's universal time, or relativistic time and for which observer? But the choice of time will be a tiny correction to the already tiny error, so it's not significant.

You are correct that these issues exist, but they aren't significant in gravitational fields as weak as the Sun's at Mercury's orbital radius.

 

[Orodruin]

##\phi## does have the same meaning, actually, so that isn't an issue. And we're looking at a rate of change of a ##\phi## coordinate with time, so ##r## isn't relevant (it appears in the derivation of the error, but we don't need to measure ##r##, just the ##\phi## where it's a minimum). It's only the ##t## that matters - are you using Newton's universal time, or relativistic time and for which observer? But the choice of time will be a tiny correction to the already tiny error, so it's not significant.

You are correct that these issues exist, but they aren't significant in gravitational fields as weak as the Sun's at Mercury's orbital radius.

Actually, ##t## doesn't matter either - you do not need to measure it. All that matters is the value of ##\phi## at the perihelion of consecutive orbits.

 

[Ibix]

Actually, ##t## doesn't matter either - you do not need to measure it. All that matters is the value of ##\phi## at the perihelion of consecutive orbits.

True - you can just look at orbit-on-orbit perihelion shift. But the OP quotes 43", which is a per-century figure. That depends on a time measurement somewhere.

 

[Orodruin]

True - you can just look at orbit-on-orbit perihelion shift. But the OP quotes 43", which is a per-century figure. That depends on a time measurement somewhere.

Well, at the level of precision, just comparing number of Earth orbits = 100 will do just fine for that. Technically of course, that is a clock.

 

[cianfa72]

As other said, ##d\Omega^2## is the Euclidean metric on a 2-sphere in polar spherical coordinates.

In the case at hand, the "center" of Schwarzschild solution is the Sun and one can evaluate the difference on ##\phi## coordinate at Mercury's successive perihelions.

 

[cianfa72]

It's only the ##t## that matters - are you using Newton's universal time, or relativistic time and for which observer?

I believe to evaluate that difference in ##\phi## on per-century we should use, in principle, the proper time here on the Earth.

 

[Vanadium 50]

The perihelion precision is a statement about orbital shape - rosette vs. ellipse. You don't need a clock to measure it.

If you insist on a clock, the effect of "which clock" is of order a million time smaller than the effect you are trying to measure.

 

[Orodruin]

precision

Precession.

You don't need a clock to measure it.

True, but as noted in #4, the value generally quoted is 43” per century. This of course translates to a number per orbit, but still.

You do however need to have enough precision to measure the precession. It is easier over longer time than over one orbit as the precession adds up.

 

[Vanadium 50]

Most of the precession, of course comes from other planets, and you need to know where (and when they are). But my point is that, sweeping that aside, a time-exposure of several orbits would be enough to show the effect - you don't need 88 days.

It's been, um, many, many orbits of Mercury since I have calculated this, but IIRC what you get is the precession per orbit (π is not π) and not per time. You need to convert.

 

[Orodruin]

The simplest approach is to compute the period of radial oscillations and compare to the time taken for the angle to increase by ##2\pi## in the small oscillation limit. This quickly gives you a good approximation.

 

[Sagittarius A-Star]

This was measured within 150 years:

Urbain Le Verrier in 1859 on the perihelion precession of Mercury
...
While of the Sun we only have meridian observations subject to great objections, we have, over the space of a century and a half, a certain number of observations of Mercury possessing great precision - I want to speak of the internal contacts of Mercury's disc with the Sun's disc, when the planet comes to pass in front of this star. As long as the place where the observation was made is well known, and as long as the astronomer had a reasonable refracting telescope and his clock was accurate to within a few seconds, the knowledge of the instant when the internal contact happened must allow us to estimate the distance between the centre of the planet and the centre of the Sun without an error of more than one arcsecond. We possess, from 1697 until 1848, twenty-one observations of this type, for which we have to be able to satisfy in the most stringent manner whether the irregularities of the movements of the Earth and of Mercury were well calculated, and if the values attributed to the perturbative masses are correct.
...
But what is remarkable is that increasing the secular movement of the perihelion by 38 arcseconds was enough to show all the observations of the transits to within one second, and even the majority of those to within one half-second. This result, so clean, which immediately shows in all comparisons a greater precision than achieved until now in astronomical theories, clearly shows the increase in the movement of the Mercury's perihelion is indispensable, and that therefore the Tables of Mercury and of the Sun have all the desired precision.

Source:
https://pappubahry.livejournal.com/572713.html