The Physics
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Opus in profectus

# Miscellaneous Units

## Discussion

### Standard values for the Earth

Earth properties, because we live here and using it for comparisons make sense.

• astronomical unit [slide]

1 au = 149,597,870,700 m exactly

• The standard distance from the Earth to the sun is a good unit for describing distances in the solar system.
• aphelion, perihelion, average
• Mercury, Jupiter, Neptune
• Pioneers, Voyagers, New Horizons

r = 6,378,100 m exactly across the equator, but also 6,356,800 m exactly pole to pole.

• Earth mass

m = 5.97217 × 1024 kg approximately (It's complcated.)

• The mass of the moon is about 0.01 m (about 1% of the Earth's mass).
• The mass of Mars is about 0.11 m (about 11% of the Earth's mass).
• The mass of Neptune is about 17 m.
• standard gravity

g = 9.80665 m/s2 exactly

Standard gravity as a unit for describing the pull of gravity itself. Great for the northern suburbs of Paris, France. Not so good for most of the the rest of the planet.

• Gravity on the surface of the Earth varies from 0.997 g on the equator to 1.003 g at the poles and has an average value of 0.999 g.
• Gravity on the Moon is about 0.17 g, often stated as ⅙ g.
• Gravity on Mars is about 0.38 g, which could roughly be approximated as ⅓ g.

Standard gravity as a unit for describing the apparent gravity felt in an accelerating reference frame.

• Objects are apparently weightless in free fall (or in orbit, which is perpetual free fall) — a 0 g reference frame.
• Riding on a roller coaster can make you feel heavier at times. Frame of reference acceleration peaks at an excitingly safe 2–3 g.
• G induced loss of consciousness (abbreviated G-LOC or g-LOC) occurs somewhere around 6–7 g for most people, but well trained pilots have been known to manage 9 g (albeit with considerable effort). Above 10 g and probably everyone will pass out. A related physiological condition is near loss or almost loss of consciousness (A-LOC)
• standard atmosphere

1 atm = 101,325 Pa exactly

• stuff you should know

### Standard values for our Sun

The sun is my favorite star. It's right next door.

• mass

m = 1.98841 × 1030kg

r = 695,700,000 m

• luminosity (power)
• Eta Carinae 5,000,000 L.
• Most massive star known R136a1 ~107 L.
• Most powerful supernova observed SN 2015L a.k.a. ASASSN-15lh 5.7 × 1011 L.
• gamma ray burst ~1018 L.
• temperature

T = 5,772 K

• irradiance (a.k.a. the solar constant, the intensity of the Sun's radiation at the Earth)

G = 1,361 W/m2

• A value that's not quite the Sun's but definitely not the Earth's.
• galactic year (cosmic year)

Symbol? = 225 million years (but I've seen 250 million as well)

• Not even close to the status of a unit right now.

### Quasi-standard values for some other astronomical objects

The universe has a lot of stuff in it. How can we make measurements of astronomical objects more meaningful?

• Jupiter properties, because so many exoplanets are similar in size to Jupiter
• mass
• Mars properties, because Mars has been occupied by space probes for so long we need to know what time it is there
• sol (Martian day)
• mean tropical year (Martian year)
• Crab nebula
• "The Crab is arguably the most exhaustively studied celestial object beyond the solar system, and the apparent stability of its luminosity at all wavelengths has made it an attractive reference for calibrating observing instruments. The brightness of other objects is often quoted in "millicrabs." But the startling observation of a powerful gamma-ray flare from the Crab last September belies that vaunted stability, and it challenges the accepted theory of how charged particles are accelerated inside supernova remnants. That theory has its beginnings in Enrico Fermi's 1949 attempt to explain the origin of cosmic rays." doi:10.1063/1.3563807
• Supernovae
• FOE for "ten to the fifty-one ergs" (1044 joules) a.k.a. a bethe for the German-Amrican nuclear physicist Hans Bethe (1906–2005).

### light year

Astronomical distances are so large that using meters is cumbersome. For really large distances the light year is the best unit. A light year is the distance that light would travel in one year in a vacuum. Since the speed of light is fast, and a year is long, the light year is a pretty good unit for astronomy. One light year is about ten petameters (ten quadrillion meters) as the following calculation shows.

Start with the definition of speed and solve it for distance. The traditional symbol for the speed of light is c from the Latin word for swiftness — celeritas.

s = ct  ⇐
 c = ∆s ∆t

 ∆s = c∆t∆s = (3.00 × 108 m/s)(365.25 × 24 × 60 × 60 s)∆s = 9.46 × 1015 mΔs ≈ 10 petameters

Since both the speed of light and the year have exactly defined values in the International System of Units, the light year can be stated with an unnecessarily large number of significant digits.

 ∆s = c∆t∆s = (299,792,458 m/s)(365.25 × 24 × 60 × 60 s)∆s = 9,460,730,472,580,800 m

Some distances in light years are provided below.

• The distance to Proxima Centauri (the star nearest the Sun) is 4.3 light years.
• The diameter of the Milky Way (the collection of stars that includes the Sun and all the stars visible to the naked eye) is about 100,000 light years.
• The distance to Andromeda (the nearest galaxy outside the Milky Way) is about 2 million light years.
• The distance to the edge of the universe (the observable part of it) is 13.77 billion light years.

### parsec

The parsec is a unit used by astronomers to describe distances from the Earth to objects outside our solar system. It a combination of the geometric words parallax and second.

Parallax is the apparent shift in the position of one object (usually one that is nearby) relative to others (usually those that are behind it) when an observer changes location (usually laterally, which is a fancy way to say sideways).

Everyone with two working eyes is familiar with this effect. Make a fist. Stick out one thumb like you are getting ready to hitchhike across America but, instead of holding your arm at your side, stick your arm out horizontally in front of you with your thumb pointing up. Look at that thumb with one eye closed. Now switch eyes. Your thumb appears to have moved relative to the stuff behind it. That's parallax.

By analogy…

• Distance is a measure of the separation between two locations.
• Time is a measure of the separation between two events.
• Angle is a measure of the difference between two directions.

The complete range of angles covering one complete rotation were traditionally divided into 360 parts called degrees (symbol °). This number 360 was probably chosen because it has lots of factors to make arithmetic easier. Also the year has around 360 days to it, so it was probably considered lucky. The degree was originally divided into small, or minute, parts called minutes (symbol ′). The minutes were then divided a second time into even smaller parts called seconds (symbol ″). 60 of these smaller units fit into the next bigger unit.

 1 rotation = 360 degrees 1 degree = 60 minutes 1 minute = 60 seconds

Or equivalently…

 1 degree = 1360 rotation 1 minute = 160 degree 121,600 rotation 1 second = 160 minute 13,600 degree 11,296,000 rotation

MORE GEOMETRY

AND I HATE THIS UNIT

Solar and planetary units Sources: When not indicated IAU, otherwise 1NIST, 2IAU, 3BIPM, 4BIPM, 5NASA. The values from NASA don't have any offical unit status, yet. * If equatorial vs. polar radius is not explicitly specified, then equatorial radius is assumed.
object characteristic value
Sun
()
luminosity 3.828 × 1026 W
temperature 5,772 K
 mass parameter 1.3271244 × 1020
m3/s2
mass1 1.98841 × 1030 kg
Earth
()
 mass parameter 3.986004 × 1014
m3/s2
mass1 5.97217 × 1024 kg
astronomical unit (au)2 149,597,870,700 m
parsec (pc)2 648,000π au
day (d) 86,400 s
Julian year (y)2 365.25 day
 light year (ly)2 9,460,730,472,580,800
m
standard gravity (g)3 9.80665 m/s2
standard atmo­sphere (atm)4 101,325 Pa
Mars
()
sol5 88,775.244 s
tropical year5 668.5921 sol
Jupiter
()
 mass parameter 1.2668653 × 1017
m3/s2
mass1 1.89812 × 1027 kg

### human constructions

• stories — typical multiunit, residential building
• 3 meters
• 10 feet (3.048 m)
• city blocks — gridded cities
• 1/20 mile (80.47 m): New York (borough of Manhattan)
• 1/16 mile (100.6 m): Houston, Milwaukee
• 1/8 mile (201.2 m): Chicago
• olympic swimming pools
• 50 × 25 × 2 m = 2500 m3 = 2,500,000 L
• football fields — International, American, Candian or Australian?
• 100 m
• 100 yards (91.44 m)
• cricket pitches — between the wickets
• 1 chain, 1/80 mile, 22 yards (20.12 m)
• bananas
• banana equivalent dose
• "Bananas produce antimatter, releasing one positron—the antimatter equivalent of an electron—about every 75 minutes. This occurs because bananas contain a small amount of potassium-40, a naturally occurring isotope of potassium. As potassium-40 decays, it occasionally spits out a positron in the process."