A working cosmogonical approximation that I've devised

A working cosmogonical approximation that I've devised

Well, couldn't resist making this up. :P
Call me greedy if you like. ;)

The Solar System is a fascinating place. Despite being one tiny fragment of the visible and observable universe, it does house its own variety of mysteries, many of which are yet to be figured out by our scientists. And recently, I've devised a small approximation for the sideral periods of the planets around the sun.

Aim

The aim of this approximation is to establish a simple arithmetic expression for calculating the sideral period of each planet around the sun, in the count of equivalent earth-days. This may not sound interesting to everyone, but I personally believe my work will catch the eye of some, at least.

The expression

Sideral period (or orbital period) is the
total time taken by a body to complete
a revolution around a second body

Before stating the expression, let me make it clear once again, that it is an empirical approximation that works moderately well for planets from Mercury to Uranus. This also includes the asteroid belt.

Ok, the expression is:

T = C * ( P + 4 ) ^ 1.5,

where T is the sideral period of the planet in equivalent earth-days,
C is a constant of proportionality, which I named Chatterjee's constant
P is the planetary constant, whose value is equivalent to the count on the series:
0 3 6 12 24 48 96 192,
where 0 is for Mercury, 3 is for Venus and so on, up and till Uranus (192)

It's to be noted that 24 here refers to the asteroid belt.

So, I named the constant after my surname. Hey, who doesn't like something like that? :D Sorry for sounding too proud of a trifle, but I just did it somewhat lightly, and it's more of a joke than anything serious. The value of C should actually be very accurately calculated, and I personally believe a better value can still be found. As of now, as per my calculations C = 11.55. 

Demonstration

Ok, so lets first calculate the sideral period of earth. We know, from observation, that it is 365.25 days. As per my expression above:

T = C * ( P + 4 ) ^ 1.5
   = 11.55 * ( 6 + 4 ) ^ 1.5
   = 11.55 * (10)^(3/2)
   = 11.55 * 31.622776602
   = 365.243069749 earth-days.

This is, as you can see, quite close to the result. In fact, it's 99.998102601% of the original value, so the error margin for earth is as low as 0.001897399%. I left the numbers in their original, long form, so that the reader can safely round it off.

Another demonstration is for Venus. We know that Venus's sideral period is 224.7 earth-days. So, using my approximation, we've: 

Kepler's third planetary law, and its graph

T = 11.55 * ( 3+ 4 ) ^ 1.5
   = 11.55 * (7)^(3/2)
   = 11.55 * 18.520259177
   = 213.723790908 days. 

This, is rather too much, I agree. The error margin here is somewhat more.


Thus, it becomes increasingly clear, that this is just an approximation and should not used as a replacement for known values.

How I arrived at this conclusion

Using the planetary sideral period chart, I initially got interested in finding out the equivalent earth-days (D). Once I'd done so, I noticed something like a proportionality arising between the sideral periods in Ds, and the number ( P + 4 ) ^ 3/2. If you've followed my post on the Science community of Google+, you know it that this particular constant has a special role to play in Titus-Bode's law. Finally, after much trial and error, I figured out the value of this constant C to be 11.54.

Theoretical basis

Originally, I started off as an empiricist, willing to use the vast repository of observed data we have gathered over the ages, thus following Brahe's principles. Later, however, I examined the result and the resulting expression a bit more seriously, and gave it a theoretical rattle. Naturally, from Kepler's third planetary law, Titus-Bode's law and a bit lengthy calculations (which I've successfully simplified using basic algebra of indices), a similar expression can be cooked up. Here's the recipe, folks.

Titus-Bode's law

In the simple form, Titus-Bode's law states the the average distance (semi-major axis) of a planet from the sun will be related to the distance of the earth from the sun, as per the expression:

d = 14960000 * ( P + 4 ) km,

The value of P is received from the series above. For example, in Jupiter's case, the result will be 15000000 * 52 = 780000000 km, which is quite close to the experimental value.

This is a much simplified form of Bode's expression, which I had previously demonstrated.

Titus-Bode's principle and its chart.
Clearly, it doesn't hold for trans-Uranian
planets (Pluto wasn't considered a
Dwarf Planet back then)

Application of Kepler's third law

Kepler's third planetary law comes packed in the mathematical expression:

t^2/d^3 = k, 

where t is the sideral period, and d is the semi-major axis (avg. distance from sun). Now:

t^2 = k * d^3,

So, t = sqrt [ k * d^3 ]

Using Earth's values, we can figure out the value of this constant k. So, since we'd like to have the answer in days in the output, lets input the time as days too. So, t(earth) = 365.25. And, and d(earth) = 149600000 km (for simplicity, lets abstain from using metres.

Therefore, t = sqrt [ { (365.25)^2 / (149600000)^3 } * (P+4)^3 * (14960000)^3 ]

= 365.25 / sqrt { (14960000)^3 * 1000 } * (P+4)^(3/2) * sqrt. (14960000)^3 

= 365.25 / sqrt (1000) * (P+4)^(1.5) 

= 11.550219154 * (P+4)^(1.5) earth-days,

Which can then be rounded off to 11.55 * (P+4)^(1.5) earth-days.

This successfully forms the theoretical basis for the derivation of my approximation.

Limitation: It holds valid only from Mercury to Uranus, and not beyond that. 

Final words 

I had approximated these calculations quite a long time ago, however my recent calculations have demonstrated the elegance of this simple principle. The primary aim of physics should to to cultivate or yield simple equations out of complicated problems. In my journey as a physics student, I've always followed this example. Ranging from Newton's F = ma (for non-relativistic motion) to Einstein's E = mc^2, we have a variety of examples to support this statement.

Using the crisp elegance of mathematical expression, the final statement is thus ready, and I can't resist typing it once again. ;)

T = C * ( P + 4 ) ^ (1.5).

NOTE: To my knowledge, there's no similar expression for this task yet. And even if there already is one, I'm not aware of it. Although I've discussed it with a few friends of mine, after a quick Google search I see that nowhere in the Internet is a similar form. Therefore, I maintain my claim on this expression, and reserve my right to call C 'Anirban's/Chatterjee's Constant', and the entire expression 'Chatterjee's Approximation/Simplification'. But definitely, it's rather a light claim, and so I believe none of you will have any objection to this fella receiving a bit of self-importance. :P 

Credits: 
Google Images, for the photos.
GIMP, for its wonderful toolset. 
Titus and Bode, for their law. 
Kepler, for his laws.