This header is the common gathering place for numerical, *machine-independent* constants in SimTK. More...
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Defines | |
Mathematical Constants | |
These are some common unitless numerical constants evaluated to 64 digits and written here in maximal (long double) precision. (These values were generated using the symbolic calculator Maple which is part of Matlab's Symbolic Toolbox.) These can be cast to lower precisions when needed, and can be used in compile-time constant expressions like 2*SimTK_PI or 1/SimTK_SQRT2 for which the compiler will properly calculate a long double result with no runtime cost. These constants are also available as type-safe, already-rounded, precision-templatized values with static memory addresses as part of our scalar system (see NTraits<T>). You should use the templatized versions when possible. The templatized versions also contain more elaborate constants such as NaN, Infinity, and "epsilon" (machine precision) which can only be generated for specific types. | |
| #define | SimTK_PI 3.141592653589793238462643383279502884197169399375105820974944592L |
| The ratio pi of a circle's circumference to its diameter in Euclidean geometry. | |
| #define | SimTK_E 2.718281828459045235360287471352662497757247093699959574966967628L |
| e, or exp(1). | |
| #define | SimTK_LN2 6.931471805599453094172321214581765680755001343602552541206800095e-1L |
| The natural (base e) logarithm of 2. | |
| #define | SimTK_LN10 2.302585092994045684017991454684364207601101488628772976033327901L |
| The natural (base e) logarithm of 10. | |
| #define | SimTK_LOG2E 1.442695040888963407359924681001892137426645954152985934135449407L |
| log2(e). | |
| #define | SimTK_LOG10E 4.342944819032518276511289189166050822943970058036665661144537832e-1L |
| log10(e). | |
| #define | SimTK_SQRT2 1.414213562373095048801688724209698078569671875376948073176679738L |
| The square root of 2. | |
| #define | SimTK_SQRT3 1.732050807568877293527446341505872366942805253810380628055806979L |
| The square root of 3. | |
| #define | SimTK_CBRT2 1.259921049894873164767210607278228350570251464701507980081975112L |
| The cube root of 2. | |
| #define | SimTK_CBRT3 1.442249570307408382321638310780109588391869253499350577546416195L |
| The cube root of 3. | |
Physical Constants | |
How to combine uncertainties (extracted from http://physics.nist.gov/cuu/Uncertainty/combination.html): Assume measured quantities are x1, y1 with u1=uncertainty(x1), u2=uncertainty(x2). We want to combine them into a new quantity y and calculate u=uncertainty(y). Addition rule: y = a1*x1 + a2*x2 for factors a1,a2. then u^2 = a1*u1^2 + a2*u2^2 Multiplication rule y = a*x1^e1*x2^e2 for factor a and exponents e1,e2. Let ur1=u1/|x1|, ur2=u2/|x2| be the relative uncertainties, ur is u/|y|. then ur^2 = e1^2*ur1^2 + e2^2*ur2^2, u = ur*|y| | |
| #define | SimTK_AVOGADROS_NUMBER 6.0221415e23L |
| Avogadro's number (NA) is defined as the number of atoms in 12g of pure Carbon-12 in its unbound, rest state. | |
| #define | SimTK_MASS_OF_PROTON_IN_MD 1.00727646688L |
| Mass of a proton in MD units. | |
| #define | SimTK_MASS_OF_NEUTRON_IN_MD 1.00866491560L |
| Mass of a neutron in MD units. | |
| #define | SimTK_MASS_OF_ELECTRON_IN_MD 5.4857990945e-4L |
| Mass of an electron in MD units. | |
| #define | SimTK_CHARGE_OF_PROTON_IN_SI 1.60217653e-19L |
| Atomic charge unit e expressed in MKS unit of Coulombs. | |
| #define | SimTK_CHARGE_OF_PROTON_IN_MD 1.L |
| Atomic charge unit e expressed in MD units, which uses e as its charge unit! The charge on an electron is just the negative of this value. | |
| #define | SimTK_MOLAR_CHARGE_IN_SI 9.6485338e+4L |
| The charge of 1 mole of protons, expressed in Coulombs. | |
| #define | SimTK_MOLAR_CHARGE_IN_MD SimTK_AVOGADROS_NUMBER |
| The charge of 1 mole of protons, expressed in MD units where the unit of charge is just the charge on one proton. | |
| #define | SimTK_LIGHTSPEED_IN_SI 2.99792458e+8L |
| Speed of light c is exact in MKS units of m/s. | |
| #define | SimTK_LIGHTSPEED_IN_MD 2.99792458e+5L |
| Speed of light c is exact in MD units of nm/ps. | |
| #define | SimTK_GRAVITATIONAL_CONSTANT_IN_SI 6.6742e-11L |
| Newton's gravitational constant G in N-m^2/kg^2 = m^3 kg^-1 s^-2. | |
| #define | SimTK_GRAVITATIONAL_CONSTANT_IN_MD 1.10827e-34L |
| Newton's gravitational constant G in (kJ/mol)-nm^2/u^2 = nm^3 u^-1 ps^-2. | |
| #define | SimTK_MAGNETIC_PERMEABILITY_IN_SI 1.256637061435917295385057353311801153678867759750042328389977837e-6L |
| Free space magnetic permeability constant mu0 in SI units (exact). | |
| #define | SimTK_MAGNETIC_PERMEABILITY_IN_MD 1.94259179e-8L |
| Free space magnetic permeability constant mu0 in MD units (not exact). | |
| #define | SimTK_ELECTRIC_PERMITTIVITY_IN_SI 8.854187817620389850536563031710750260608370166599449808102417149e-12L |
| Free space permittivity constant e0 = 1/(mu0*c^2) Farad/m = Coulomb^2/(N-m^2) (exact in SI units). | |
| #define | SimTK_ELECTRIC_PERMITTIVITY_IN_MD 5.7276575e-4L |
| Free space permittivity constant e0=1/(mu0*c^2) e^2/(kN-nm^2) using MD permeability and MD lightspeed. | |
| #define | SimTK_COULOMB_CONSTANT_IN_SI 8.9875517873681764e+9L |
| Coulomb's constant kappa = 1/(4pi*e0)=1e-7*c^2 N-m^2/Coulomb^2 (exact in SI units). | |
| #define | SimTK_COULOMB_CONSTANT_IN_MD 1.38935456e+2L |
| Coulomb's constant kappa = 1/(4*pi*e0) in MD units. | |
| #define | SimTK_COULOMB_CONSTANT_IN_KCAL_ANGSTROM 3.32063711e+2L |
| Coulomb's constant kappa = 1/(4*pi*e0) in kcal-Angstroms/e^2. | |
| #define | SimTK_MOLAR_GAS_CONSTANT_SI 8.314472L |
| This is the gas constant R in (J/mol)/K. | |
| #define | SimTK_MOLAR_GAS_CONSTANT_MD 8.314472e-3L |
| This is the gas constant R in (kJ/mol)/K. | |
| #define | SimTK_MOLAR_GAS_CONSTANT_KCAL_ANGSTROM 1.9872065e-3L |
| This is the gas constant R in (kcal/mol)/K. | |
| #define | SimTK_BOLTZMANN_CONSTANT_SI 1.3806505e-23L |
| Boltzmann's constant in SI units of joules/kelvin; just divide R by NA. | |
| #define | SimTK_BOLTZMANN_CONSTANT_MD SimTK_MOLAR_GAS_CONSTANT_MD |
| Boltzmann's constant in MD units of (kJ/mol)/kelvin; same as R. | |
| #define | SimTK_BOLTZMANN_CONSTANT_KCAL_ANGSTROM SimTK_MOLAR_GAS_CONSTANT_KCAL_ANGSTROM |
| Boltzmann's constant in Kcal-Angstrom units of (kcal/mol)/kelvin; same as R. | |
Unit Conversion Factors | |
In each case here, given a value in the units mentioned first in the name, you should multiply by the given constant to produce the equivalent quantity measured in the units that appear second in the name. You can perform the reverse conversion by dividing by the constant, or by using another conversion constant with the names reversed if one is supplied here. | |
| #define | SimTK_RADIAN_TO_DEGREE 5.729577951308232087679815481410517033240547246656432154916024386e+1L |
| Convert radians to degrees. | |
| #define | SimTK_DEGREE_TO_RADIAN 1.745329251994329576923690768488612713442871888541725456097191440e-2L |
| Convert degrees to radians. | |
| #define | SimTK_KCAL_TO_KJOULE 4.184L |
| Convert Kcal to Kjoule (also Kcal/mol to Kjoule/mol). | |
| #define | SimTK_KJOULE_TO_KCAL 2.390057361376673040152963671128107074569789674952198852772466539e-1L |
| Convert Kjoule to Kcal (also Kjoule/mol to Kcal/mol). | |
| #define | SimTK_DALTON_TO_GRAM 1.66053886e-24L |
| Convert atomic mass unit (amu, Dalton) to g. | |
| #define | SimTK_E_TO_COULOMB SimTK_CHARGE_OF_PROTON_IN_SI |
| Convert proton charge units to Coulombs. | |
| #define | SimTK_EV_TO_JOULE SimTK_CHARGE_OF_PROTON_IN_SI |
| Convert electron volts to Joules. | |
This header is the common gathering place for numerical, *machine-independent* constants in SimTK.
These include unitless mathematical constants like Pi, as well as physical constants and unit conversion factors. This is NOT the place for computational, machine-dependent constants like NaN, Infinity, number of digits in a float, etc. Those belong in templatized classes in the manner of the C++ std::numeric_limits<T> class.
These constants are provided at extremely high precision as compile-time macros in long double precision. By using very high precision we ensure sufficient accuracy for any IEEE long double precision implementation (they can be 64, 80, or 128 bits). These constants can be used as raw material for providing nicer templatized constants in appropriate precisions and unit systems.
Units: our most common unit systems are the "SI" (MKS) system, and the "MD" system used for molecular dynamics. SI units are meters, kg, seconds, coulombs (ampere-s), kelvins and moles. MD units are nanometers, atomic mass units (Daltons, g/mol), picoseconds, proton charge e, kelvins, and moles. Many molecular dynamicists and chemists prefer kcals for energy and angstroms for length. This does not constitute a consistent set of units, however, so we provide for it by conversion from the MD units, which are consistent. (By consistent, we mean that force units = mass-length/time^2, so f=ma!)
This file is self-contained and can be included in ANSI C programs as well as C++.
Unit systems
SI (MKS) MD KCAL-ANGSTROM --------- -------------- ------------------------ ------------------ length meter nanometer angstrom (Å) mass kg amu, dalton amu, dalton time second picosecond picosecond charge coulomb e, proton charge e, proton charge temp. kelvin kelvin kelvin substance mole mole mole
velocity m/s km/s (nm/ps) 100m/s (Å/ps)
energy J (kg-m^2/s^2) kJ/mol kcal/mol
(Da-nm^2/ps^2) (418.4 Da-Å^2/ps^2)
force N (kg-m/s^2) kJ/(mol-nm) = TN/mol kcal/(mol-Å)
(Da-nm/ps^2) (T=10^12) (418.4 Da-Å/ps^2)We always keep angles in radians internally, which are unitless. However, most humans prefer degrees where 1 degree = Pi/180 radians so we provide convenient conversions.
| #define SimTK_AVOGADROS_NUMBER 6.0221415e23L |
Avogadro's number (NA) is defined as the number of atoms in 12g of pure Carbon-12 in its unbound, rest state.
The number is 1 mole (mol).
| #define SimTK_BOLTZMANN_CONSTANT_KCAL_ANGSTROM SimTK_MOLAR_GAS_CONSTANT_KCAL_ANGSTROM |
Boltzmann's constant in Kcal-Angstrom units of (kcal/mol)/kelvin; same as R.
| #define SimTK_BOLTZMANN_CONSTANT_MD SimTK_MOLAR_GAS_CONSTANT_MD |
Boltzmann's constant in MD units of (kJ/mol)/kelvin; same as R.
| #define SimTK_BOLTZMANN_CONSTANT_SI 1.3806505e-23L |
Boltzmann's constant in SI units of joules/kelvin; just divide R by NA.
| #define SimTK_CBRT2 1.259921049894873164767210607278228350570251464701507980081975112L |
The cube root of 2.
| #define SimTK_CBRT3 1.442249570307408382321638310780109588391869253499350577546416195L |
The cube root of 3.
| #define SimTK_CHARGE_OF_PROTON_IN_MD 1.L |
Atomic charge unit e expressed in MD units, which uses e as its charge unit! The charge on an electron is just the negative of this value.
| #define SimTK_CHARGE_OF_PROTON_IN_SI 1.60217653e-19L |
Atomic charge unit e expressed in MKS unit of Coulombs.
The charge on an electron is just the negative of this value.
| #define SimTK_COULOMB_CONSTANT_IN_KCAL_ANGSTROM 3.32063711e+2L |
Coulomb's constant kappa = 1/(4*pi*e0) in kcal-Angstroms/e^2.
This is the constant that appears in Coulomb's law f(r)= kappa*q1*q2/r^2. This is an exact conversion from MD units (which are inexact).
| #define SimTK_COULOMB_CONSTANT_IN_MD 1.38935456e+2L |
Coulomb's constant kappa = 1/(4*pi*e0) in MD units.
This is the constant that appears in Coulomb's law f(r)= kappa*q1*q2/r^2.
Coulomb's consant in MD units uses MD e0 & c:
1/(4*pi*e0)=1e5*1.60217653e-19^2*6.0221415e23*c^2 kN-nm^2/e^2 (=kJ-nm/e^2)
(exact in SI units but not exact in MD)
| #define SimTK_COULOMB_CONSTANT_IN_SI 8.9875517873681764e+9L |
Coulomb's constant kappa = 1/(4pi*e0)=1e-7*c^2 N-m^2/Coulomb^2 (exact in SI units).
This is the constant that appears in Coulomb's law f(r)= kappa*q1*q2/r^2.
| #define SimTK_DALTON_TO_GRAM 1.66053886e-24L |
Convert atomic mass unit (amu, Dalton) to g.
This is 1/NA (NA=avogadro's number).
| #define SimTK_DEGREE_TO_RADIAN 1.745329251994329576923690768488612713442871888541725456097191440e-2L |
Convert degrees to radians.
| #define SimTK_E 2.718281828459045235360287471352662497757247093699959574966967628L |
e, or exp(1).
| #define SimTK_E_TO_COULOMB SimTK_CHARGE_OF_PROTON_IN_SI |
Convert proton charge units to Coulombs.
This is the same as the conversion from electron volts to Joules, and both are just the charge of a proton in SI units.
| #define SimTK_ELECTRIC_PERMITTIVITY_IN_MD 5.7276575e-4L |
Free space permittivity constant e0=1/(mu0*c^2) e^2/(kN-nm^2) using MD permeability and MD lightspeed.
| #define SimTK_ELECTRIC_PERMITTIVITY_IN_SI 8.854187817620389850536563031710750260608370166599449808102417149e-12L |
Free space permittivity constant e0 = 1/(mu0*c^2) Farad/m = Coulomb^2/(N-m^2) (exact in SI units).
| #define SimTK_EV_TO_JOULE SimTK_CHARGE_OF_PROTON_IN_SI |
Convert electron volts to Joules.
This is the same as the conversion from proton charge units to Coulombs, and both are just the charge of a proton in SI units.
| #define SimTK_GRAVITATIONAL_CONSTANT_IN_MD 1.10827e-34L |
Newton's gravitational constant G in (kJ/mol)-nm^2/u^2 = nm^3 u^-1 ps^-2.
Conversion is (nm/m)^3 (u/kg)^-1 (ps/s)^-2
= 1.66053886(28)e-24L (uncertainty: 28e-32)
This is why gravity doesn't matter in molecular systems. Don't try this in single precision -- you'll run out of exponent!
| #define SimTK_GRAVITATIONAL_CONSTANT_IN_SI 6.6742e-11L |
Newton's gravitational constant G in N-m^2/kg^2 = m^3 kg^-1 s^-2.
The force between two point masses m1,m2 separated by a distance d is
F = -G m1*m2/d^2
(with the "-" indicating an attractive force).
| #define SimTK_KCAL_TO_KJOULE 4.184L |
Convert Kcal to Kjoule (also Kcal/mol to Kjoule/mol).
| #define SimTK_KJOULE_TO_KCAL 2.390057361376673040152963671128107074569789674952198852772466539e-1L |
Convert Kjoule to Kcal (also Kjoule/mol to Kcal/mol).
| #define SimTK_LIGHTSPEED_IN_MD 2.99792458e+5L |
| #define SimTK_LIGHTSPEED_IN_SI 2.99792458e+8L |
| #define SimTK_LN10 2.302585092994045684017991454684364207601101488628772976033327901L |
| #define SimTK_LN2 6.931471805599453094172321214581765680755001343602552541206800095e-1L |
| #define SimTK_LOG10E 4.342944819032518276511289189166050822943970058036665661144537832e-1L |
log10(e).
| #define SimTK_LOG2E 1.442695040888963407359924681001892137426645954152985934135449407L |
log2(e).
| #define SimTK_MAGNETIC_PERMEABILITY_IN_MD 1.94259179e-8L |
Free space magnetic permeability constant mu0 in MD units (not exact).
Convert kg->g/mole, m->nm, C->e = (4*pi*1e5)*1.60217653e-19^2*6.0221415e23
(exact in SI units, but not exact here)
| #define SimTK_MAGNETIC_PERMEABILITY_IN_SI 1.256637061435917295385057353311801153678867759750042328389977837e-6L |
Free space magnetic permeability constant mu0 in SI units (exact).
= 4*pi * 1e-7 exactly in N/A^2 (Newtons/Ampere^2) = kg-m/C^2
| #define SimTK_MASS_OF_ELECTRON_IN_MD 5.4857990945e-4L |
| #define SimTK_MASS_OF_NEUTRON_IN_MD 1.00866491560L |
| #define SimTK_MASS_OF_PROTON_IN_MD 1.00727646688L |
Mass of a proton in MD units.
The atomic mass unit u (or amu) is defined as 1/12 of the mass of a Carbon-12 atom, unbound and in its rest state. This definition matched to Avogadro's number's definition ensures that 1 mole of particles of mass 1u each has total mass exactly 1g. This is synonymous with the dalton (Da), with units of g/mole, so 1u = 1Dalton = 1g/mole. We will use Da for this mass unit, with kDa being a common mass measure for large biomolecules.
| #define SimTK_MOLAR_CHARGE_IN_MD SimTK_AVOGADROS_NUMBER |
The charge of 1 mole of protons, expressed in MD units where the unit of charge is just the charge on one proton.
So in MD units this is just Avogadro's number.
| #define SimTK_MOLAR_CHARGE_IN_SI 9.6485338e+4L |
The charge of 1 mole of protons, expressed in Coulombs.
1.60217653(14)e-19 C/e * 6.0221415(10)e23 = 9.6485338(18)e+4
| #define SimTK_MOLAR_GAS_CONSTANT_KCAL_ANGSTROM 1.9872065e-3L |
This is the gas constant R in (kcal/mol)/K.
This is an exact conversion from MD units, differing only in the use of kcal here vs. kJ in MD.
| #define SimTK_MOLAR_GAS_CONSTANT_MD 8.314472e-3L |
This is the gas constant R in (kJ/mol)/K.
This is an exact conversion from SI units, differing only in the use of kJ here vs. J in SI.
| #define SimTK_MOLAR_GAS_CONSTANT_SI 8.314472L |
This is the gas constant R in (J/mol)/K.
| #define SimTK_PI 3.141592653589793238462643383279502884197169399375105820974944592L |
The ratio pi of a circle's circumference to its diameter in Euclidean geometry.
| #define SimTK_RADIAN_TO_DEGREE 5.729577951308232087679815481410517033240547246656432154916024386e+1L |
Convert radians to degrees.
| #define SimTK_SQRT2 1.414213562373095048801688724209698078569671875376948073176679738L |
The square root of 2.
| #define SimTK_SQRT3 1.732050807568877293527446341505872366942805253810380628055806979L |
The square root of 3.
1.6.1