Monday, June 22, 2020

[ceefdxbt] Mercury perihelion clock

The contribution of general relativity to the perihelion precession of Mercury is 42.98 arc-seconds per 36525 days (i.e., per Julian century).  Create a model, a device or animation, which illustrates this.

Mercury's elliptic orbit is eccentric enough that its deviation from a circle is perceptible to the eye.  The sun, at one focus, is located very noticeably not at the center of the ellipse.  Therefore, when the ellipse of our model rotates (precesses) around a fixed sun, the fact that it has rotated will be noticeable.

In reality, the GR contribution to precession results in an (additional) full precession revolution every 3.0 million gregorian (Earth) years.  Mercury's orbital period is 87.97 days, so this is 12.5 million Mercury orbits.

For a model, easiest is to accelerate time, increasing Mercury's speed.

Consider having Mercury circle the sun once a minute, perhaps repurposing the second hand of a clock.  This corresponds to accelerating time by a factor of 130000.  Then, 12.5 million orbits takes 23.8 years.

Consider Mercury circling the sun 10 times a second, 10 Hz.  This is probably about as fast as the eye can perceive movement, i.e., can even tell which direction Mercury is orbiting.  This corresponds to accelerating time by a factor of 78 million.  Then, perihelion precession due to GR takes 14.5 days to complete a full revolution.  Or, tweak it so that the precession period is one lunar month (lunation), perhaps repurposing some existing mechanism for displaying the phase of the moon.  This corresponds to accelerating time by a factor of 38 million, and Mercury's orbital frequency would be 4.907 Hz.

If we don't care about being able to perceive Mercury moving in the model, we could accelerate time more.  If the ellipse makes a full rotation every 12 hours, perhaps repurposing the hour hand of a clock, then this corresponds to accelerating time by a factor of 2.2e9.

The contribution of general relativity to Mercury's perihelion precession is relatively small compared to the contribution of everything else to the precession, most notably the contribution of the other planets.  To illustrate (or celebrate) general relativity, we have to ignore the rest of Mercury's perihelion precession.  The models above depicting a precessing ellipse therefore will not reflect Mercury's actual motion.

Mercury's total perihelion precession is 574 arcsec per julian century relative to the ICRF (of which general relativity contributes 43 arcsec per julian century), or 5600 arcseconds per julian century relative to earth (of which general relativity still contributes 43 arcsec per julian century).  I don't know what "relative to earth" means.  It might mean relative to the location of the vernal equinox.  100/((5600-574)/60/60/360) = 25785.4 julian year = 25786.5 gregorian year, which is close to the period of earth's axial precession.

Also, in reality, the eccentricity of Mercury's orbit changes over time as well.  (How does it vary?)

How was the contribution to perihelion precession from other planets, a N-body problem with Newtonian gravity, precisely calculated long before computers?  How did astronomers know that Newton's theory of gravity did not match observations?

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