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Physical Units by Mark Lawrence 
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We're used to measuring things with a lot of different types of units. In the US, we measure small things in inches, medium things in feet, and large things in miles. We weigh things in pounds. Time is measured in seconds, minutes, and hours. Some things are measured in combinations, like miles per hour. And, there's all sorts of units that don't play into our daily lives very often for things like forces, power, torque, etc. In physics, we deal with all of these units on a routine basis, and more.
Most of these units were chosen for historical reasons, many of which have since been forgotten. Feet were originally the length of a man's foot. Seconds were probably chosen by the average heart rate of a man. A minute is the time it takes for the sun or the moon to rise or set. Miles were originally 1000 paces of a Roman centurion. So, this is all very interesting for walking around Europe or watching sunsets, but it really doesn't have much to do with the fundamental laws of physics.
It's purely an accident of chemistry and biology that we see time and space as so different. Using the speed of light, c, as a conversion constant, we can measure distance in seconds or time in feet. One light second is about 186,000 miles. One foot of time is about one nanosecond. So, we agree to set c equal to 1, and we agree that we will use the same measure for time and space.
Einstein's special theory of relativity tells us that E = Mc^{2}. But, we have just agreed that c is one, so energy and mass have the same units, pounds or kilograms or horsepowerhours, we can choose whatever we wish.
From quantum mechanics we learn that everything oscillates with a frequency η = E / h, where h is Planck's constant. This is completely fundamental  all energy oscillates, whether it's a photon or a bowling ball. It's just an accident of history and human perception that we choose to measure energy or mass with a different scale than time or distance. So, we'll agree to measure energy and mass using units of frequency, meaning "per second" or "per meter." Now, having made this agreement, Planck's constant is 1. We still haven't chosen a basic unit, but we have reduced most everything to this one unit.
Distance is now measured in meters. Velocity, distance per time, is meters per meter, so velocity has no dimensions. In our system of measurement, a velocity of one is the speed of light. The speed limit on most freeways is 65 miles per hour which equals about c / 10,000,000. If physicists were running the highways, apparently highway signs would say "Speed limit 10^{7}." That's it, no units.
Acceleration is velocity per second, so acceleration has dimensions of "per meter." Mass also has dimensions of "per meter," so F = Ma tells us that force has dimensions of "per meter^{2}." Gauss' law of static electricity tells us that F = e^{2 } / r^{2}, so e, the electric charge, is dimensionless. The fine structure constant α = e ^{2}/4π is also dimensionless.
Finally, we have one more fundamental unit in nature: G, Newton's gravitational constant. The units of G can be deduced from Newton's equation of gravity, F = GmM / r^{2}. Since the force has dimensions of "per meter^{2}," and the 1/r^{ 2} term has dimensions of "per meter^{2}," we see that GmM has no dimension. Therefore, since Mm has dimensions of "per meter^{2}," G must have dimensions of "meter^{2}." Now, our big leap: we'll set G to one, and therefore start using the same dimensions that the universe naturally uses  we'll call them "natural dimensions," sometimes referred to as "God's units."
In cgs (centimetergramseconds) units, G = 2/3*10^{7} cm^{3} / gs^{2}. Thus, we see that G / c^{3 } = 1/4 * 10^{38} s/g. Now, we multiply by h = 6.6*10^{27} ergsec = 6.6*10^{27} gcm^{2 }/sec and we get 1.63 * 10^{65 }cm^{2}. Finally, the square root of this number is √(hG/c^{3}) = 4.04*10^{33} cm. This will be our fundamental unit of length and time, which we will call the Planck, abbreviated as P. We can live with just one fundamental unit, but for convenience sake we will define one additional unit. Our mass and energy unit will be h / (c * Planck) = √(hc/G) = 5.45*10^{5} g, which will call the Stone, abbreviated as E. Note that the Stone is simply 1 / Planck. Wherever we use Stone, we could write Planck^{1}.
Now we're left with just a few factors of 2π and such in various places. For example, our Lagrangian is now in units of mass, as we expect for an energy term, so the time integral of the Lagrangian, the action, is dimensionless, as we expect. We'll use the action to find the phase of a particle as phase = exp( i 2π S t ). We could have scaled our units to eliminate this 2π, but we prefer to leave it in as an explicit reminder of the difference between time and radians.
Below is a conversion table and a list of constants. This is enough in most cases to work real problems in natural units and get answers in MKS.
Symbol  Name  Natural  cgs  MKS 
m  Mass  1 stone  5.45*10^{5} grams  5.45*10^{8} kilograms 
l  Length  1 planck  4.037*10^{33} centimeters  4.037*10^{35} meters 
l  Length  1 planck  4.037*10^{25} Ångstrom  4.265*10^{51} lightyears 
t  Time  1 planck  1.346*10^{43} seconds  3.74*10^{47} hours 
t  Time  1 planck  1.558*10^{48} days  4.265*10^{51} years 
E  Energy  1 stone  4.9*10^{16} ergs  4.9*10^{9} joules 
E  Energy  1 stone  3.06*10^{22} MeV  3.55*10^{32} °K 
V  Volume  1 planck^{3}  6.58*10^{98} cm^{3}  6.58*10^{104} meters^{3} 
v  Velocity  1  3*10^{10} cm/second  3*10^{8} meters/second 
a  Acceleration  1 stone  2.23*10^{53} cm/second^{2}  2.23*10^{51} meters/second^{2} 
a  Acceleration  1 stone  2.27*10^{50} g  
a  Tidal Acceleration  1 stone^{2}  5.52*10^{85} /second^{2}  5.62*10^{84} g/meter 
F  Force  1 stone^{2}  4.9*10^{17} dynes  8.22*10^{45} Newtons 
p  Pressure  1 stone^{4}  3.38*10^{79} dynes/cm^{2}  3.38*10^{70} Newtons/meter^{2} 
d  Mass Density  1 stone^{4}  8.28*10^{92} gm/cm^{3}  8.28*10^{83} kg/meter^{3} 
Constant  MKS value  Natural value 
G  6.673*10^{11} N m^{2} / kg^{2}  1 stone^{2} 
e  1.602*10^{19} C  .303 
c  3*10^{10} cm / sec  1 
h  6.62607544*10^{34} J s  1 
hc  1.9856*10^{23} kg m^{3} / s^{2}  1 
Boltzman's constant k  1.38*10^{16} ergs / °K  2.82*10^{33} stone / °K 
m_{e} electron mass  9.1096*10^{31} kg = .511 MeV  1.67*10^{23} stone 
m_{p} proton mass  1.6725*10^{27} kg = 938.3 MeV  3.066*10^{20} stone 
m_{n} neutron mass  1.6748*10^{27} kg = 939.6 MeV  3.07*10^{20} stone 
Compton wavelength h / 2π m_{e} c  3.86*10^{13} m = 3.86*10^{3} Å  9.56*10^{21} planck 
Bohr radius h^{2} / 4π^{2} m_{e} e^{2}  5.29*10^{11} m = .529 Å  1.31*10^{24} planck 
Rydberg constant ½ m_{e} c^{2} α^{2}  13.6 eV  4.44*10^{28} stone 
mass of sun  1.987*10^{30} kg  3.646*10^{37} stone 
mass of earth  5.97*10^{24} kg  1.095*10^{32} stone 
mass of moon  7.32*10^{22} kg  1.343*10^{30} stone 
radius of sun  6.96*10^{8} m  1.724*10^{43} planck 
radius of earth  6371 km  1.578*10^{41} planck 
radius of moon  1.7375*10^{6} m  4.304*10^{40} planck 
mean orbital radius of earth  1 AU = 1.495*10^{11} m  3.703*10^{45} planck 
mean orbital radius of moon  3.844*10^{8} m  9.522*10^{42} planck 
year  3.156*10^{7} s  2.345*10^{50} planck 
g earth  9.8 m / s^{2} = G M / r^{2}  4.35*10^{51} stone 
g sun  273.4 m / s^{2} = G M / r^{2}  1.23*10^{49} stone 
Schwarzchild radius of earth  8.85 mm = 2GM / c^{2}  2.19*10^{32} planck = 2*M_{earth} 
Schwarzchild radius of sun  2944 m = 2GM / c^{2}  7.292*10^{37} planck 
The following data is from NASA/JPL
Planet  Mean radius km  Mass x10^{23} kg  Density g/cc  Rotation Period Sidereal h  Gravity m s^{2}  Escape Velocity km/s  Orbit Period Sidereal yr  Orbit Radius AU  Eccentricity 
Mercury  2440  3.30  5.4  1408  3.7  4.4  0.241  0.3871  0.2056 
Venus  6052  48.7  5.2  5832  8.87  10.4  0.615  0.7233  0.0068 
Earth  6371  59.7  5.5  23.93  9.78  11.2  1.00  1.0000  0.0167 
Mars  3390  6.42  3.9  24.6  3.69  5.03  1.88  1.5237  0.0934 
Jupiter  69911  19000  1.3  9.92  23.1  59.5  11.86  5.2034  0.0484 
Saturn  58232  5680  0.69  10.66  8.96  35.5  29.45  9.5371  0.0542 
Uranus  25362  868  1.3  17.24  8.69  21.3  84.02  19.191  0.0472 
Neptune  24624  1020  1.6  16.11  11.0  23.5  164.8  30.069  0.0086 
Pluto  1151  .131  2.0  153  0.655  1.3  248  39.482  0.2488 
Satellite  Mean Radius Km  Mass x10^{20} kg  Density g/cc  Period Days  Orbit Radius Km  Eccentricity 
Earth  
Moon  1737.5  732  3.34  27.322  384400.  0.0554 
Mars  
Phobos  11.1  0.107E3  1.87  0.319  9380.  0.0151 
Deimos  6.2  22.4E6  2.25  1.262  23460.  0.0002 
Jupiter  
Io  1827  890  3.53  1.624  421800.  0.0041 
Europa  1561  478  3.01  3.551  671100.  0.0094 
Ganymede  2631  1476  1.94  7.155  1070400.  0.0011 
Callisto  2410  1072  1.834  16.69  1882700.  0.0074 
Saturn  
Mimas  198.6  0.381  1.17  0.942  185600.  0.0206 
Enceladus  249.4  1.04  1.60  1.370  238100.  0.0001 
Tethys  529.9  6.15  0.991  1.888  294700.  0.0001 
Dione  559.  10.9  1.49  2.737  377400.  0.0002 
Rhea  764.  23.1  1.24  4.518  527100.  0.0009 
Titan  2575.  1340.  1.88  15.95  1221900.  0.0288 
Iapetus  718.  19.4  1.25  79.33  3560800.  0.0284 
Uranus  
Ariel  578.9  13.4  1.67  2.520  190900.  0.0012 
Umbriel  584.7  11.6  1.40  4.144  266000.  0.0039 
Titania  788.9  35.1  1.71  8.706  436300.  0.0011 
Oberon  761.4  30.0  1.63  13.46  583500.  0.0014 
Miranda  235.8  0.657  1.20  1.413  129900.  0.0013 
Neptune  
Triton  1353  213  2.06  5.877  354800.  0.0000 
Nereid  170.  0.308  1.5  360.14  5513400.  0.7512 
Proteus  210.  0.502  1.3  1.122  117647.  0.0005 
Pluto  
Charon  593.  16.1108.  1.853  6.387  19410.  0.0002 