Centimetre-Gram-Second (CGS) System¶

Overview¶

The Centimetre-Gram-Second (CGS) system is a metric system of physical units based on the centimetre, gram, and second as base units. It was widely used before the adoption of the SI system and is still employed in certain scientific fields, particularly in theoretical physics, astronomy, and some areas of chemistry.

Historical Context¶

The CGS system was introduced in the 1870s and became the dominant metric system in science for many decades. While it has been largely superseded by the SI for most applications, it remains popular in:

Astrophysics and cosmology: Many astronomical calculations and constants
Electrodynamics: Gaussian and electrostatic CGS variants
Plasma physics: Natural unit scales align with CGS
Physical chemistry: Some molecular and atomic calculations

Base Units¶

The CGS system uses three base units, all derived from SI units in mp-units:

namespace mp_units::cgs {

inline constexpr auto centimetre = si::centi<si::metre>;   // 10⁻² m
inline constexpr auto gram = si::gram;                     // 10⁻³ kg
inline constexpr auto second = si::second;                 // 1 s

}

Base Quantity	CGS Unit	Symbol	Relation to SI
length	centimetre	cm	1 cm = 0.01 m
mass	gram	g	1 g = 0.001 kg
time	second	s	1 s = 1 s

Base Unit: Gram vs Kilogram

Unlike SI, where kilogram is the base unit, CGS uses gram as its base mass unit. This makes the CGS system more consistent with its length scale (centimetre) for everyday laboratory measurements.

Derived Units¶

The CGS system defines several derived units with special names, particularly in mechanics:

Quantity	CGS Unit	Symbol	Definition	SI Equivalent
acceleration	gal	Gal	cm/s²	0.01 m/s²
force	dyne	dyn	g·cm/s²	10⁻⁵ N
energy/work	erg	erg	dyn·cm	10⁻⁷ J
pressure	barye	Ba	dyn/cm²	0.1 Pa
dynamic viscosity	poise	P	g/(cm·s)	0.1 Pa·s
kinematic viscosity	stokes	St	cm²/s	10⁻⁴ m²/s
wavenumber	kayser	K	cm⁻¹	100 m⁻¹

namespace mp_units::cgs {

inline constexpr struct gal final : named_unit<"Gal", centimetre / square(second)> {} gal;
inline constexpr struct dyne final : named_unit<"dyn", gram * centimetre / square(second)> {} dyne;
inline constexpr struct erg final : named_unit<"erg", dyne * centimetre> {} erg;
inline constexpr struct barye final : named_unit<"Ba", gram / (centimetre * square(second))> {} barye;
inline constexpr struct poise final : named_unit<"P", gram / (centimetre * second)> {} poise;
inline constexpr struct stokes final : named_unit<"St", square(centimetre) / second> {} stokes;
inline constexpr struct kayser final : named_unit<"K", one / centimetre> {} kayser;

}

Unit Symbols¶

Convenient unit symbols are provided for common usage:

namespace mp_units::cgs::unit_symbols {

inline constexpr auto cm = centimetre;
inline constexpr auto g = gram;
inline constexpr auto s = second;
inline constexpr auto Gal = gal;
inline constexpr auto dyn = dyne;
inline constexpr auto Ba = barye;
inline constexpr auto P = poise;
inline constexpr auto St = stokes;
inline constexpr auto K = kayser;

// Commonly used squared and cubic units
inline constexpr auto cm2 = square(centimetre);
inline constexpr auto cm3 = cubic(centimetre);
inline constexpr auto s2 = square(second);
inline constexpr auto s3 = cubic(second);

}

Usage Examples¶

Basic Mechanics¶

using namespace mp_units;
using namespace mp_units::cgs::unit_symbols;

// Force calculation
quantity mass = 500 * g;
quantity acceleration = 980 * cm / s2;
quantity force = mass * acceleration;
std::cout << force.in(dyn) << "\n";      // 490000 dyn

// Energy calculation
quantity distance = 10 * cm;
quantity work = force * distance;
std::cout << work.in(cgs::erg) << "\n";  // 4900000 erg

Gravitational Acceleration¶

The gal unit is named after Galileo and is commonly used in gravimetry:

using namespace mp_units;
using namespace mp_units::cgs::unit_symbols;

// Standard gravity in CGS
quantity g_earth = 980.665 * Gal;                 // 980.665 cm/s²

// Gravity variation measurements (microgal precision)
quantity gravity_anomaly = 50 * si::micro<Gal>;   // 50 μGal

Viscosity Measurements¶

using namespace mp_units;
using namespace mp_units::cgs::unit_symbols;

// Dynamic viscosity (poise)
quantity water_viscosity = 0.01 * P;          // Water at 20°C: 0.01 P = 1 cP

// Kinematic viscosity (stokes)
quantity water_kin_viscosity = 0.01 * St;     // Water at 20°C: 0.01 St = 1 cSt

Spectroscopy¶

The kayser (reciprocal centimetre) is used in spectroscopy for wavenumbers:

using namespace mp_units;
using namespace mp_units::cgs::unit_symbols;

// Wavenumber of a spectral line
quantity wavenumber = 15000 * K;                           // 15000 cm⁻¹

// Convert to wavelength
quantity wavelength = 1. / wavenumber;
std::cout << wavelength.in(si::nano<si::metre>) << "\n";   // 666.667 nm

Converting Between CGS and SI¶

mp-units handles conversions automatically:

using namespace mp_units;

// CGS to SI
quantity force_cgs = 1000. * cgs::dyne;
quantity force_si = force_cgs.in(si::newton);              // 0.00001 N

quantity energy_cgs = 1e7 * cgs::erg;
quantity energy_si = energy_cgs.in(si::joule);             // 1 J

// SI to CGS
quantity force_newton = 1 * si::newton;
quantity force_dyne = force_newton.in(cgs::dyne);          // 100000 dyn

quantity pressure_pascal = 100 * si::pascal;
quantity pressure_barye = pressure_pascal.in(cgs::barye);  // 1000 Ba

Mixed Calculations¶

using namespace mp_units;
using namespace mp_units::si::unit_symbols;

// Start with SI
quantity velocity_si = 10 * m / s;

// Convert to CGS for calculation
quantity velocity_cgs = velocity_si.in(cgs::centimetre / si::second);  // 1000 cm/s

// Work with CGS units
quantity mass_cgs = 50 * cgs::gram;
quantity momentum = mass_cgs * velocity_cgs;               // 50000 g·cm/s

// Convert result back to SI
quantity momentum_si = momentum.in<double>(kg * m / s);    // 0.5 kg·m/s

CGS Variants¶

Electromagnetic Units

The CGS system has several variants for electromagnetic quantities:

Gaussian CGS: Uses statcoulomb and statvolt (no separate electrical base unit)
ESU (Electrostatic): Based on Coulomb's law
EMU (Electromagnetic): Based on Ampère's force law

The mp-units library currently provides only the mechanical CGS units shown above. Electromagnetic CGS variants may be added in future versions.

Advantages and Limitations¶

Advantages¶

Natural scale: CGS units align well with laboratory measurements
Simpler formulas: Some physics formulas are simpler (e.g., ε₀ = μ₀ = 1 in Gaussian CGS)
Historical compatibility: Allows working with older scientific literature
Convenient magnitudes: Many atomic and molecular quantities are order-of-unity

Limitations¶

Not standard: Largely replaced by SI in modern usage
Electromagnetic confusion: Multiple incompatible variants for E&M quantities
Limited adoption: Fewer tools and standards support CGS

Conversion Safety with mp-units

While manual CGS-to-SI conversions historically required careful tracking of powers of 10 (e.g., 1 erg = 10⁻⁷ J, 1 dyne = 10⁻⁵ N), mp-units handles all conversions automatically and safely. The library ensures dimensional correctness and applies the correct conversion factors, eliminating this traditional source of errors.

When to Use CGS¶

Consider using CGS units when:

Working with astrophysical calculations where CGS is standard
Reading or reproducing results from older scientific papers
Interfacing with legacy code or data in CGS units
Working in fields where CGS remains conventional (e.g., plasma physics)

For general engineering and most modern scientific work, SI units are recommended.