Mentor

## The time it takes the Earth to go around the Sun.

What is the precise diameter of a lumpy Russet potato?

You are asking for the precise value of a quantity that is known to vary. Does that make any sense?
 Ask Gordon Ramsey, he’ll know that one. Do you know how much the Sidereal year varies D H? According to NASA it now is 365.246007919206 days. See post 32. What fixed stars are used as a reference for the Earth to complete one orbit? They did use the star Sirius until they found it had proper motion.
 Mentor I don't know where you got that value of 365.246. I suspect that your value of 365.246 day resulted from using something other than 365.25 days per (Julian) year as a conversion factor. I see values of 1.0000174 years (this page: http://ssd.jpl.nasa.gov/?planet_phys_par) or, when I use the site to generate an ephemeris for the Earth, I see values of 1.0000174 years or 365.25636 days. 1.0000174 years and 365.25636 days are the same number. Multiply 1.0000174 by 365.25 and you'll get 365.25635535, or 365.25636. Astronomers use the Julian year (exactly 365.25 days of exactly 86,400 seconds each) to measure time. JPL does not use those values to compute their ephemerides. Those values for the sidereal period are presented for informational purposes only. Here's what they use instead: JPL starts with initial estimates of the masses of the bodies that comprise the solar system combined with initial estimates of the positions and velocities of those bodies at some key point in time, the epoch time. They next use numerical integration techniques to propagate these initial estimates forwards and backwards in time according to the laws of physics. They then develop a set of Chebyshev polynomial-based lookup tables from this forward and backward propagation. These Chebyshev polynomials let them quickly compute the states (position and velocity) of any modeled celestial body at some given time. Next they use this lookup table to compute what an observation of some body at a given point in time would yield. They have lots (lots and lots) of real observations on hand. Comparing the observed values with the predicted ones gives them an idea of how good / how bad these lookup tables are, and also gives them an idea of how to improve them. So they improve them by tweaking those initial estimates of mass, epoch position, and epoch velocity. Then they repeat the process, and keep repeating until the observed vs prediction errors become sufficiently small. When you use the Horizons system to generate an ephemeris, you are using the end result of this huge numerical grinding process.
 I use this ephemeris. Ephemeris Type ELEMENTS. Target Body Earth Moon Barycenter [EMB] [3]. Center Sun [Bodycenter]. Time Span Start=2013-03-04 Stop=1 d. Table settings default. Display Output default. Formatted HTML. http://ssd.jpl.nasa.gov/horizons.cgi#results This ephemeris gives the date of this year’s perihelion as, Julian date 2456295.7720, 03/01/2013 06:31:45, was this correct? How much does the Sidereal year vary by, not this much surly? What star or point in space did they start all these calculation from this year?
 Mentor Those are osculating orbital elements. They are computed from position and velocity by assuming that the orbit is Keplerian. This Keplerian assumption makes for a (somewhat) easy set of transformations from position and velocity to orbital elements. The problem is that the orbit is not Keplerian. The Earth-Moon barycenter doesn't follow an Keplerian ellipse. It's close to elliptical, but that "close to" means that some funky results will arise.
 Now you’ve mentioned ellipses. How far does the Earth travel around the Sun each year? 365.25636004 days * Earths average velocity 29784.813 m/s = 939.9536 million kilometres. Or? (360 degrees / 365.25636004 days) * 939.9536 million kilometres = 939.89623 million kilometres

 Quote by D H You made another error somewhere in your computation of the Earth's gravitational parameter. That number is just wrong. You also made a mistake in multiplying the mass of the Sun by the gravitational constant. The standard gravitational parameters are observables. The mass of the Sun is computed by dividing the solar gravitational constant by the universal gravitational constant. Use the wrong value of G (e.g., the value provided by google) and you'll get a wrong value. Use inconsistent values of the solar mass and G (e.g., the values provided by wikipedia) and you'll get a wrong answer. You'll also get a wrong answer if you use 1.00000261 au (value provided by wikipedia) as the length of the Earth's semi-major axis. That value is wrong. If you use 149597887.5 km as the semi-major axis length and a combined Sun+Earth+Moon gravitational parameter of 132712440018+398600.4418+4902.7779 km3/s2, you'll get a value for the period that is within 3.8 seconds of the sidereal year (*not* anomalistic year): http://www.wolframalpha.com/input/?i...+sidereal+year. That's about as good as you're going to get with this simplistic formula that ignores the effects of the other planets and that ignores relativistic effects.
I'm curious then -- what's the "next-more-complicated but as simple as possible" formula or method that would allow one to get to just within sub-second accuracy? Also, how much accuracy does one need to demand before one has to start getting into General Relativity?

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 Quote by Jimmy B Now you’ve mentioned ellipses. How far does the Earth travel around the Sun each year? 365.25636004 days * Earths average velocity 29784.813 m/s = 939.9536 million kilometres. Or? (360 degrees / 365.25636004 days) * 939.9536 million kilometres = 939.89623 million kilometres
Is that really your "average" velocity? Or is it what the Earth's speed would be if the Earth were in a circular orbit?

Most people that ask about the Earth's average velocity (or the average velocity of any other object) are really asking for a ballpark figure and it's common to just give the circular velocity for objects that have low eccentricities. Calculating the actual "average" velocity is hard to do, so most people don't bother. (The difficulty arises in calculating the circumference of an ellipse - it becomes even more difficult if you're calculating the circumference of a perturbed ellipse.)

In other words, if you really do care (why when it's not used for anything?), make sure your source actually calculated the average velocity.
 What is the Earths average velocity around the Sun? I just plucked that value of the internet. Each web site that I went on had a slightly different average velocity. I was spoilt for choice. They also had their own method of calculating it.
 This is one of many. $$vp=\sqrt((1+e)\mu/(1-e)a)$$ $$va=\sqrt((1-e)\mu/(1+e)a)$$ vp = velocity at perihelion and va = velocity at aphelion. vp = 30.25375649 km and va =29.32291236 km mean velocity = vp+va/2 I got a mean velocity of 29.7883344272490 km. have I done it right? Is it correct?
 Mentor No. You have *way* too many digits, for one thing. For another, how are you defining "mean"? Suppose you take a two hour drive. You drive a steady 30 MPH for the first hour, then 60 MPH for the second. What's your average speed?
 Thanks for the reply, I’ll discount that one. Here’s another. Average speed of the Earth around the Sun = (2*pi)*(a/p) = 29784.8132 m/s. a = 1.49598E+11 Earths semi-major axis, and p = 31558149.76 seconds in a sidereal year. Earths orbital perimeter = 365.256363007 days in a Sidereal year * 29784.8132 average speed = 9.39954E+11 meters (939.9536 million kilometers). Is this any better? Will not be long now until the Vernal Equinox, according to US Naval Observatory it will be on the 20/03/2013 at 11:02 UTC. Is this correct?

 Quote by sshai45 I'm curious then -- what's the "next-more-complicated but as simple as possible" formula or method that would allow one to get to just within sub-second accuracy? Also, how much accuracy does one need to demand before one has to start getting into General Relativity?
Same here, if you ever find out please let me know.
 Hi, all. I’m back with another question. If you want to find the Earths mean anomaly using this equation “M = n t”. What value should “n” be, the daily motion, if “t = 0”? Should it be 360/Sidereal year, 360/Tropical year, 360/Anomalistic year, or is it none of these?
 Wikipedia has the mean anomaly of 357.51716 degrees, is this correct? If it is, the only way I can satisfy this equation “M = Mo + nt” that is on Wikipedia is, Mo = 357.5176 degrees, n = 0.98560025850 degrees, and “t” = 365.259644606150 days. “t” has a very near value of a anomalistic year. Is the anomalistic year used for this equation?
 I also got M = 6.2832173611570 radians (360.001836557620 degrees) using this equation from wiki. M = n t = sqrt ( G M / a^3) * t My values were, G M = 1.33E+11, a^3 = 149598261.0 km^3, t = 31558432.541760 seconds in a anomalistic year. There does seem to be a relationship. Getting somewhere at last; who knows I might be able to answer one of my grandson question after-all this.

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