Computing the period of a meteor using two ways.

[solarblast]

Computing the period of a meteor using two ways.

I'm looking at code that computes the orbit found for a meteor. It computes the period of the meteor as:

a = 13.07 # semi-major axis in AU
P = m.sqrt(a*a*a) # From the program

which yields P = 47.2512586393 Years

If I use the "standard" approach

P = 2*pi*m.sqrt((a*a*a)/mu),

where mu (=GM) is the standard gravitational constant (G*M or GM).
3986004418.0 in km**3/sec**2,

then I get:

P = 703462.683141 # units?

They certainly don't look close. A source I'm looking at says 4*pi*pi/G = 1, but they are talking about solar masses.

So why the large discrepancy? Units?

 

[D H]

It is always a good idea to use the right constants and consistent units.

You are using the gravitational parameter for the Earth, not the Sun. The solar gravitational parameter is 1.327124400180×10¹¹ km³/s². To have consistent units, you need to either convert that 13.07 AU to kilometers or convert the gravitational parameter to AU³/year².

With the first technique, convert that 13.07 AU to 1.955244×10⁹ km. Then
$T = 2\pi\sqrt{a^3/\mu_{\odot}} \approx 1.49116\times10^9$ seconds, or 47.25 sidereal years ( = 47.28 years).

With the second technique, it's very handy to fold that factor of 2*pi into the square root. With this, the sun's gravitational parameter divided by 4*pi^2 is $u_{\odot}' = 0.999997054$ AU³/sidereal year². Note that this is numerically equal to one to six places. This is what let's you get away with using [itex]T=\sqrt{a³}[/itex].

 

[solarblast]

It is always a good idea to use the right constants and consistent units.

You are using the gravitational parameter for the Earth, not the Sun. The solar gravitational parameter is 1.327124400180×10¹¹ km³/s². To have consistent units, you need to either convert that 13.07 AU to kilometers or convert the gravitational parameter to AU³/year².

With the first technique, convert that 13.07 AU to 1.955244×10⁹ km. Then
$T = 2\pi\sqrt{a^3/\mu_{\odot}} \approx 1.49116\times10^9$ seconds, or 47.25 sidereal years ( = 47.28 years).

With the second technique, it's very handy to fold that factor of 2*pi into the square root. With this, the sun's gravitational parameter divided by 4*pi^2 is $u_{\odot}' = 0.999997054$ AU³/sidereal year². Note that this is numerically equal to one to six places. This is what let's you get away with using [itex]T=\sqrt{a³}[/itex].

They certainly don't look close. A source I'm looking at says 4*pi*pi/G = 1, but they are talking about solar masses.

So why the large discrepancy? Units?[/QUOTE]

Well, your certainly right about the units and the earth. I was using material I found on the web, and often it was unit-less. The book the computer program on never mentions how to compute P, but it's certainly in the code.

Thanks very much.

 

[solarblast]

Well, not all worked out. Something is wrong. Here's a Python program and output.

import math as m

GMsun = 13271244018.0 # km**3/sec**-2
Gsun = 6.67834*(10**-11)

GMearth = 398600.4418 # km**3/sec**-2
Rearth = 6378140.0 # radius of Earth in km
SunToEarth = 149597870.7 #km
EarthMass = GMearth = 3986004418.0 # km**3/sec**-2
EarthMass = 5.9742*10**24

sidereal_year = 31558149.8 # sec
AU = 149598000.0 # KM per AU

pi = 3.14152965


#
print "Meteor Period"

mu = GMsun/(4*pi*pi)

a = 13.07*AU
a1 =13.07
P = (2.0*pi*m.sqrt((a*a*a)/mu)) / sidereal_year # Web
Pbook = m.sqrt(a1*a1*a1) # Wray book
print "P: ", P, " and Book: ",Pbook

print "Check for mu_sun: ", mu

=================
Meteor Period
P: 938.808534723 and Book: 47.2512586393 <---P from code is way off
Check for mu_sun: 336178021.818

 

[solarblast]

To put a close to this, I have no idea why P = (2.0*pi*m.sqrt((a*a*a)/mu)) is useful. As it happens p^2/a^3 = 1 for all objects in orbit around the sun, planets anyway. So once you have a, the period is determine by the sqrt of a^3.

 

[D H]

To put a close to this, I have no idea why P = (2.0*pi*m.sqrt((a*a*a)/mu)) is useful. As it happens p^2/a^3 = 1 for all objects in orbit around the sun, planets anyway. So once you have a, the period is determine by the sqrt of a^3.

The expression T^2 = A^3 is valid only for orbits around the Sun, only for objects much less massive than the Sun, and only if you express time in sidereal years, distance in AU, and only if you don't care that the result is only good to a few decimal places.

 

[solarblast]

And the value of the other equation is what?

 

[D H]

The other formula can be used to calculate the period of a satellite's orbit about the Earth. Or about Jupiter. Or the period of an exoplanet about some other star. And so on.

 

[solarblast]

Ah, thanks. Why is that true though?