Asking a question here is quite intimidating for me -- while I love Physics, my high-school understanding only allows me to go so far... I've been trying to solve this problem for the last couple of days without success, so I thought that perhaps someone here wouldn't mind helping me to better understand it.
Lets say, our origin location is London, defined by the WGS84 coordinates 51.507222
and -0.1275
. What I want to do, is calculate the magnitude (?) and direction of the gravitic force exerted by the Sun
and Moon
bodies (only, for the sake of simplicity) on the origin location for a given date. Using pyEphem:
origin = ephem.Observer()
origin.lon, origin.lat = '51.507222', '-0.1275'
origin.date, origin.pressure = '2012/05/27 23:00:00', 0 # 0 ignores atmosferic refraction
moon = ephem.Moon(origin)
sun = ephem.Sun(origin)
print moon.distance * ephem.meters_per_au # 388989920.985 m (from Earth)
print sun.distance * ephem.meters_per_au # 1.51606987131e+11 m (from Earth)
Assuming $1.988435 * 10^{30} kg$, $5.9721986 * 10^{24} kg$ and $7.3459 * 10^{22} kg$ as the mass of the Sun, Earth and Moon, respectively, I'm now able to calculate the gravitational force for each body individually:
$$ F(Earth|Sun) = G {5.9721986 * 10^{24} * 1.988435 * 10^{30} \over (1.51606987131 * 10^{11})^2} = 3.448 * 10^{22} N $$
$$ F(Earth|Moon) = G {5.9721986 * 10^{24} * 7.3459 * 10^{22} \over (3.88989920985 * 10^{8})^2} = 1.935 * 10^{20} N $$
Combining Gravitational Force Vectors in a 1-D Plane
Now, consider the following scenarios ($\#1$ and $\#2$) in the image, where the circles represent the Sun, Moon and Earth and the little black square represents a (small) object of $100$ kg at our origin location:
PS: Sorry for the imagery but I don't know how to adequately explain this using the proper jargon...
I'm not completely sure if this is the case, but my intuition tells me that the object will be slightly heavier in the first scenario and slightly lighter in the other. However, if I try to work out the math:
$$ F(Earth|Obj) = G {5.9721986 * 10^{24} * 100 \over (6.3675 * 10^{6})^2} = 983 N $$
$$ F_{\#1} = F(Earth|Obj) + F(Earth|Moon) + F(Earth|Sun) = 3.467 * 10^{22} N $$
$$ F_{\#2} = F(Earth|Obj) - F(Earth|Moon) - F(Earth|Sun) = -3.467 * 10^{22} N $$
Clearly, I'm doing something wrong here. I suspect the right answer is:
$$ F(Sun|Obj) = G {1.988435 * 10^{30} * 100 \over (1.51606987131 * 10^{11})^2} = 0.5773 N $$
$$ F(Moon|Obj) = G {7.3459 * 10^{22} * 100 \over (3.88989920985 * 10^{8})^2} = 0.00324 N $$
$$ F_{\#1} = F(Earth|Obj) + F(Moon|Obj) + F(Sun|Obj) = 983.58054 N $$
$$ F_{\#2} = F(Earth|Obj) - F(Moon|Obj) - F(Sun|Obj) = 982.41946 N $$
Which looks way more realistic, the mass difference on the object would only be of $≈ ±59$ grams. However, I'm a bit confused, since the mass of the object is part of the mass of the Earth, and this Wikipedia entry states that: "large spherically symmetrical masses attract and are attracted as if all their mass were concentrated at their centers" and "r is the distance between the centers of the masses".
Combined Gravitational Force Vectors in a Equatorial Coordinate System?
Continuing with the pyEphem code, I can get the Apparent Topocentric Position of both the Sun and the Moon, as seen from our chosen date and origin location (pressure = 0 ignores atmospheric refraction):
print moon.ra, moon.dec # 9:43:33.65, 7:47:56.6
print sun.ra, sun.dec # 4:20:51.51, 21:28:53.9
Converting these values to degrees yields:
- Moon: $145.89º$ Right Ascension, $+7.80º$ Declination
- Sun: $65.21º$ Right Ascension, $+21.48º$ Declination
Where the Right Ascension is $Φ$ (azimuthal) and the Declination is $Θ$ (polar); I believe $R$ (radial) should either be the distance from the origin location to the center of mass $X$ (but then I wouldn't be able to calculate the resultant force, would I?) or the result of $F(Body|Obj)$ -- would this make sense?
So, converting the above spherical coordinates into $(X, Y, Z)$ Cartesian coordinates gives:
$$ X_{Earth} = F(Earth|Obj) * cos(0) * sin(0) = 0 $$
$$ Y_{Earth} = F(Earth|Obj) * sin(0) * sin(0) = 0 $$
$$ Z_{Earth} = F(Earth|Obj) * cos(0) = 983 $$
$$ X_{Sun} = F(Sun|Obj) * cos(21.48) * sin(65.21) = 0.48802 $$
$$ Y_{Sun} = F(Sun|Obj) * sin(21.48) * sin(65.21) = 0.19185 $$
$$ Z_{Sun} = F(Sun|Obj) * cos(65.21) = 0.24189 $$
$$ X_{Moon} = F(Moon|Obj) * cos(7.80) * sin(145.89) = 0.00180 $$
$$ Y_{Moon} = F(Moon|Obj) * sin(7.80) * sin(145.89) = 0.00025 $$
$$ Z_{Moon} = F(Moon|Obj) * cos(145.89) = -0.00268 $$
$$ (0, 0, 983) + (0.48802, 0.19185, 0.24189) + (0.00180, 0.00025, -0.00268) $$
Using the above vectors as input on WolframAlpha yields a vector length of $983.239$ (Newtons?) -- is this my resultant force? This would mean that my $100$ kg object has gained (or lost?) $262$ grams, which is more than my exaggerated 1-dimensional scenario would allow. What might I be doing wrong?
I realize that this probably isn't the right way to tackle this, I also noticed that there is a vector equation for Newton's Law of Universal Gravitation, but honestly I don't even understand the notation it uses.
I hope this isn't too much of a dumb question... And please let me know if I should explain it better.