Constants and Metric System

In many scientific computing problems, numbers are not abstract but reflect the realistic meanings. In other words, these numbers only make sense on top of a well-defined metric system.

What Is a Metric System

For example, when we talk about the distance between two objects, I write down a number 30, but what does 30 mean in reality? Is it meters, kilometers, miles, or lightyears? Another example, what is the speed of light? Well, this is really depends on what metrics you are using, e.g., km/s, m/s, mile/h … Things can get really messy in computation if we do not unify the metric system in a numerical library. The translation between different metrics is often important in real-world application. I do not intend to dig deep into the metric system here, so please read online articles to find out more, e.g., Wiki: Outline of the metric system.

Four Metric Systems

There are four metrics adopted in Owl, and all of them are wrapped in the Owl.Const module.

Const.SI: International System of Units
Const.MKS: MKS system of units
Const.CGS: Centimetre–gram–second system of units
Const.CGSM: Electromagnetic System of Units

All the metrics defined in these four systems can be found in the interface file owl_const.mli.

In general, SI is much newer and recommended to use. International System of Units (French: Système international d’unités, SI), historically also called the MKSA system of units for metre–kilogram–second–ampere. The SI system of units extends the MKS system and has 7 base units, by expressing any measurement of physical quantities using fundamental units of Length, Mass, Time, Electric Current, Thermodynamic Temperature, Amount of substance and Luminous Intensity, which are Metre, Kilogram, Second, Ampere, Kelvin, Mole and Candela respectively. Here is a nice one-page poster from NPL to summarise what have talked about SI.

Units of measurement in the SI metric system

SI Prefix

As a computer scientist, you must be familiar with prefixes such as kilo, mega, giga. SI system includes the definition of these prefixes as well. But be careful (especially for computer science guys), the base is 10 instead of 2. These prefixes are defined in the Const.Prefix module.

  Const.Prefix.peta;;
  Const.Prefix.tera;;
  Const.Prefix.giga;;
  Const.Prefix.mega;;
  Const.Prefix.kilo;;
  Const.Prefix.hecto;;

Example: Physics and Math constants

Now we can safely talk about the distance between two objects, light of speed, and a lot of other real-world stuff with atop of a well-defined metric system in Owl. See the following examples.


  Const.SI.light_year;;     (* light year in SI system *)
  Const.MKS.light_year;;    (* light year in MKS system *)
  Const.CGS.light_year;;    (* light year in CGS system *)
  Const.CGSM.light_year;;   (* light year in CGSM system *)

How about Planck’s constant?


  Const.SI.plancks_constant_h;;     (* in SI system *)
  Const.MKS.plancks_constant_h;;    (* in MKS system *)
  Const.CGS.plancks_constant_h;;    (* in CGS system *)
  Const.CGSM.plancks_constant_h;;   (* in CGSM system *)

The table below shows some physical constants that the SI module includes:

Physical constants {#tbl:constant:physic}
Constant name	Explanation
`speed_of_light`	speed of light in vacuum
`gravitational_constant`	Newtonian constant of gravitation
`plancks_constant_h`	Planck constant
`plancks_constant_hbar`	reduced Planck constant
`astronomical_unit`	one astronomical unit in meters
`light_year`	one light year in meters
`parsec`	one light year in meters
`grav_accel`	standard acceleration of gravity
`electron_volt`	electron volt
`mass_electron`	electron mass
`mass_muon`	muon mass
`mass_proton`	proton mass
`mass_neutron`	neutron mass
`rydberg`	Rydberg constant
`boltzmann`	Boltzmann constant
`molar_gas`	molar gas constant
`standard_gas_volume`	molar volume of ideal gas (273.15 K, 100 kPa)
`bohr_radius`	Bohr radius
`stefan_boltzmann_constant`	Stefan-Boltzmann constant
`thomson_cross_section`	Thomson cross section in square metre
`bohr_magneton`	Bohr magneton in Joules per Tesla
`nuclear_magneton`	Nuclear magneton in Joules per Tesla
`electron_magnetic_moment`	electron magnetic moment in Joules per Tesla
`proton_magnetic_moment`	proton magnetic moment in Joules per Tesla
`faraday`	Faraday constant
`electron_charge`	electron volt in Joules
`vacuum_permittivity`	vacuum electric permittivity
`vacuum_permeability`	vacuum magnetic permeability
`debye`	one debye in coulomb metre
`gauss`	one gauss in maxwell per square metre

Some basic mathematical constants are also provided in Owl, though some constants in advanced mathematics are not yet included such as the golden ratio or Euler–Mascheroni constant.

Math constants {#tbl:constant:math}
Constant name	Explanation
`pi`	Pi
`e`	Natural constant
`euler`	Euler constant

Besides these constants, we also provide some frequently used computations based on them, including:

log2e (\(\log_2 e\))
log10e (\(\log_10 e\))
loge2 (\(\log_e 2\))
loge10 (\(\log_e 10\))
logepi (\(\log_e \pi\))
pi2 (\(2\pi\))
pi4 (\(4\pi\))
pi_2 (\(\pi / 2\))
pi_4 (\(\pi / 4\))
sqrt1_2 (\(\sqrt{\frac{1}{2}}\))
sqrt2 (\(\sqrt{2}\))
sqrt3 (\(\sqrt{3}\))
sqrtpi (\(\sqrt{\pi}\))

International System of Units

Now that you know how to use constants, we will use the International System of Units (SI) module as an example to show the constants we include in Owl. These units are all derived from the seven basic units we have mentioned, and can be categorised according to different application fields.

Time

The base SI unit for time measurement is second.

Time units {#tbl:constant:time}
Constant name	Explanation
`minute`	one minute in seconds
`hour`	one hour in seconds
`day`	one day in seconds
`week`	one week in seconds

Length

The base SI unit for length measurement is metre.

Length units {#tbl:constant:length}
Constant name	Explanation
`inch`	one inch in metres
`foot`	one foot in metres
`yard`	one yard in metres
`mile`	one mile in metres
`mil`	one mil in metres
`fathom`	one fathom in metres
`point`	one point in metres
`micron`	one micron in metres
`angstrom`	one angstrom in metres
`nautical_mile`	one nautical mile in metres

Area

Measuring area and volume still relies on SI base unit metre.

Area units {#tbl:constant:area}
Constant name	Explanation
`hectare`	one hectare in square meters
`acre`	one acre in square meters
`barn`	one barn in square meters

Volume

Volume units {#tbl:constant:volume}
Constant name	Explanation
`liter`	one liter in cubic meters
`us_gallon`	one gallon (US) in cubic meters
`uk_gallon`	one gallon (UK) in cubic meters
`canadian_gallon`	one Canadian gallon in cubic meters
`quart`	one quart (US) in cubic meters
`cup`	one cup (US) in cubic meters
`pint`	one pint in cubic meters
`fluid_ounce`	one fluid ounce (US) in cubic meters
`tablespoon`	one tablespoon in cubic meters

Speed

The base units for speed are that of time and length.

Speed units {#tbl:constant:speed}
Constant name	Explanation
`miles_per_hour`	miles per hour in metres per second
`kilometers_per_hour`	kilometres per hour in metres per second
`knot`	one knot in metres per second

Mass

The base unit for presenting mass is kilogram (kg).

Mass units {#tbl:constant:mass}
Constant name	Explanation
`pound_mass`	one pound (avoirdupous) in kg
`ounce_mass`	one ounce in kg
`metric_ton`	1000 kg
`ton`	one short ton in kg
`uk_ton`	one long ton in kg
`troy_ounce`	one Troy ounce in kg
`carat`	one carat in kg
`unified_atomic_mass`	atomic mass constant
`solar_mass`	one solar mass in kg

Force

Measuring force relies on the SI derived unit: newton, and one newton equals to 1 kilogram metre per squared second.

Force units {#tbl:constant:force}
Constant name	Explanation
`newton`	SI derived unit (\(kg \cdot m \cdot s^{-2}\))
`gram_force`	one gram force in newtons
`kilogram_force`	one kilogram force in newtons
`pound_force`	one pound force in newtons
`poundal`	one poundal in newtons
`dyne`	one dyne in newtons

Energy

The unit of measuring energy level is joule, which equals to one kilogram square metre per square second.

Energy units {#tbl:constant:energy}
Constant name	Explanation
`joule`	SI base unit
`calorie`	one calorie (thermochemical) in Joules
`btu`	one British thermal unit (International Steam Table) in Joules
`therm`	one therm (US) in Joules
`erg`	one erg in Joules

Power

The unit of power is watts, a SI derived unit. One watts equals to one kilogram square metre per cubic second, or one Joule per second.

Power units {#tbl:constant:power}
Constant name	Explanation
`horsepower`	one horsepower in watts

Pressure

To measure pressure we often use pascal as a standard unit. One pascal equals to a kilogram per metre per square second, or a newton per square metre.

Pressure units {#tbl:constant:pressure}
Constant name	Explanation
`bar`	one bar in pascals
`std_atmosphere`	standard atmosphere in pascals
`torr`	one torr (mmHg) in pascals
`meter_of_mercury`	one metre of mercury in pascals
`inch_of_mercury`	one inch of mercury in pascals
`inch_of_water`	one inch of water in pascals
`psi`	one psi in pascals

Viscosity

The poise is a unit in dynamic viscosity and the stokes is for kinematic viscosity. They are actually included in the CGS-based system for electrostatic units.

Viscosity units {#tbl:constant:viscosity}
Constant name	Explanation
`poise`	base unit
`stokes`	base unit

Luminance

Candela is the base unit for luminance, and both lumen and lux are derived units.

Luminance units {#tbl:constant:illuminatti}
Constant name	Explanation
`stilb`	Candela per square metre
`lumen`	luminous flux, Candela square radian, SI derived unit
`phot`	base unit
`lux`	one lux in phots, SI derived unit
`footcandle`	one footcandle in phots
`lambert`	base unit
`footlambert`	one footlambert in lambert

Radioactivity

The SI unit of radioactivity is becquerel, named in honour of the scientist Henri Becquerel, defined as one transformation (or decay or disintegration) per second. The other base units such as ampere, second, and kilogram are also used.

Radioactivity units {#tbl:constant:radioactivity}
Constant name	Explanation
`curie`	one curie in becquerel
`roentgen`	one ampere second per kilogram
`rad`	erg per gram

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