Thursday, April 22, 2010

Earth Day - Earth Units

In honor of Earth day, I thought I would take a look at what it would mean to do physics in 'Earth' units. What do I mean by that? Well lets be anti-Copernican here, in fact lets assume the opposite of the
Copernican principle, and state that the Earth is privileged in the universe and define all of our units around the Earth.

So, I will put a little subscript earth on all of the 'earth' units. They are to be read as 'earth meters' or 'earth amps', etc. We will take as our starting point the mass, radius and day of the earth, normalizing all of our standards to that. This gives us our initial conversion factors
\[ 1 g_{\oplus} = M_{\oplus} = 5.9742 \times 10^{34} \text{ kg} \]
\[ 1 m_{\oplus} = R_{\oplus} = 6378.1 \text{ km} \]
\[ 1 s_{\oplus} = T_{\oplus} = 86,400 \text{ s} \]

From this, we can figure out what all of the other 'earth' units would be.


Lengths


Some things we are used to talking about start to look a little simpler in these units. The surface area of the earth would be
\[ 4 \pi\ m_{\oplus}^2 \sim 12.6\ m_{\oplus}^2 \]
and the volume of the earth would be
\[ \frac{4\pi}{3}\ m_{\oplus}^3 \sim 4.2\ m_{\oplus}^3 \]

One earth velocity would be
\[ 1\ \frac{ m_{\oplus} }{ s_{\oplus} } = 73.8\ \frac{ m }{ s} \]

so that the velocity of a person standing at the equator would be about
\[ 2 \pi\ \frac{ m_{\oplus }}{s_{\oplus} } \sim 6.3\ \frac{ m_{\oplus } }{ s_{\oplus } } \]

and one earth acceleration would be
\[ 1\ \frac{ m_{\oplus} }{ s_{\oplus}^2 } = 8.5 \times 10^{-4} \frac{ m}{s^2} \]
so that the gravitational acceleration we feel on the earth in earth accelerations would be
\[ g \sim 1.15 \times 10^4\ \frac{m_{\oplus}}{s_{\oplus}^2 } \]
which is a little more awkward than the 10 that it is in SI.

After this all of the numbers start to get pretty silly.

Mechanics


One earth energy is
\[ 1\ J_{\oplus} = 3.3 \times 10^{28}\ J \]
and earth force
\[ 1\ N_{\oplus} = 5.1\times 10^{21}\ N \]
the gravitational constant becomes
\[ G = 3.88 \times 10^{-25} \ \frac{m_{\oplus}^3 }{ g_{\oplus} s_{\oplus}^2} \]
earth power
\[ 1\ W_{\oplus} = 3.8 \times 10^{23} \ W \]
earth pressure
\[ 1\ Pa_{\oplus} = 1.2 \times 10^8 \ Pa \]


Electrical and Thermal

Additionally if I take the Boltzmann constant and electrical constant as fundamental dimensionfull quantities, and set them equal to 1 (i.e. CGS-type Earth units), I can use them to discover earth units dealing with electrical or thermal phenomenon.

An earth kelvin is
\[ 1\ K_{\oplus} = 3.2 \times 10^{28} \ K \]
earth coulomb
\[ 1\ C_{\oplus} = 4.6 \times 10^{17} \ C \]
earth amp
\[ 1\ A_{\oplus} = 5.3 \times 10^{12} \ A \]
an earth volt
\[ 1\ V_{\oplus} = 7.1 \times 10^{10} \ V \]
an earth farad
\[ 1\ F_{\oplus} = 6.4\times 10^6 \ F \]
an earth ohm
\[ 1\ \Omega_{\oplus} = 14 m\Omega\]
a earth henry
\[ 1\ H_{\oplus} = 1170 \ H \]
an earth electric field
\[ 1\ \frac{V_{\oplus}}{m_{\oplus}} = 11200\ \frac{V}{m} \]
an earth tesla
\[ 1\ T_{\oplus} = 152 \ T \]

etc....


Lessons

So, it looks like if we really decide to fly in the face of the Copernican principle and look to the earth as something fundamental in the universe, these considerations can suggest a bunch of other relevant values for other kinds of dimensionfull quantities in the world. If the earth really was something super special in the universe, and if somehow its design was intimately connected with the properties of physics at large, then all of these different values ought to have some kind of deep meaning.

Unfortunately, as far as I can tell, they are just a bunch of random numbers. Looks like the Copernican principle wins again. Nobody should tell the earth. Its feelings might get hurt.

2 comments:

  1. The luminosity of the sun is almost exactly 1000 W_ⴲ.

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  2. Well put. Using more precise numbers (i.e. not rounding to the second decimal place), I get
    \[ L_{\odot} = 1019 W_{\oplus} \]
    although I think its more a coincidence than saying anything deep.

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