This series is intended to revisit some of the deeper, more thoughtful questions directed at me during my time serving on All Experts. Each question will be given, followed by the answer.
Question: I am puzzled by
Einstein’s conclusion that Mercury’s advance of perihelion is 43 arc seconds
per year. How can that be obtained from the equation he used in The
Principle of Relativity?
Diagram showing (exaggerated) advance of Mercury's perihelion.
Ans. What
Einstein is really concerned with here is the amount of rotation of 'the
planetary ellipse' due to the effects of gravitation in general relativity. In this case the ellipse is that of the planet
Mercury.
The equation used and purported to show the amount of this rotation (i.e. advance of perihelion) is given as:
e = [24 (p)3 a2]/ T2 c2 [1 - e2]
where e is the advance (or rotation) in seconds of arc, T is the period of revolution in seconds, c the velocity of light and e the eccentricity of the orbit.
Einstein, on page 164, states that for Mercury e= 43 seconds per century
The equation used and purported to show the amount of this rotation (i.e. advance of perihelion) is given as:
e = [24 (p)3 a2]/ T2 c2 [1 - e2]
where e is the advance (or rotation) in seconds of arc, T is the period of revolution in seconds, c the velocity of light and e the eccentricity of the orbit.
Einstein, on page 164, states that for Mercury e= 43 seconds per century
The
speed of light c = 3 x 1010 cm/s in the text.
In this context, 1 astronomical unit (AU) = 1.5 x 1013 cm. But from a Table of distances, Mercury's semi-major axis = 0.387 AU or:
a = (1.5 x 1013 cm) (0.387) = 5.8 x 1012 cm
The period, T (in seconds) is just the length of Mercury's year (in days = 87.96, again from a Table) multiplied by the seconds-length of an Earth day, or 86,400 s, hence::
T = 7.6 x 106 s
The eccentricity, e from a similar Table is e = 0.205.
Substituting all these values into the given equation yields:
e = 5.036 x 10-7 radian
To get the equivalent seconds of arc (or arcsec) we use 1 rad (radian) = 57.3 degree where one degree has 3600 seconds. Thus, 1 radian will have:
2.063 x 105 arcsec
So, the associated arcsec for e will be:
(5.036 x 10- 7 rad) x (2.063 x 105 arcsec/ rad) =
0.104 arcsec
We are still not finished because the quantity is defined per CENTURY
At this point, you need to recall the PERIOD of Mercury is 0.2405 YRS.
So, the number of arcsec of perihelion advance per Earth years is:
0.104 arcsec/ 0.2405 years = 0.432
and over 100 years:
eta = 100 yr x (0.432 arcsec/ yr) ~ 43.2 arcsec or near to what Einstein noted on the page.
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