A unit is a string of one or more fields separated
by 'space' or '.' (to indicate multiply) or '/' (to indicate divide).
Multiple separators are acted upon (i.e. m//s == m.s).
Separators are acted upon left-to-right (i.e. m/s/A == (m/s)/A;
use () to indicate otherwise (e.g. m/(s/A) )).
A field is a name, or a unit enclosed in (), optionally followed by an,
optionally signed, decimal constant. E.g. m.(m/s)-2 == m-1.s2 )
Note that a 'space' or '.' before an opening '(' can be omitted.
A name can consist of case-sensitive letters, '_', ''', ':', '"' and '0'
('0' not as first character). Digits 1-9 are allowed if preceded with
an '_'. Possible legal names are e.g. Jy, R0, R_1, "_2.
A name must be defined in a Unit map before it can be used.
All SI units and some customary units are part of the classes. User
defined names can be added by the UnitMap::putUser() function (see
UnitMap). A special set of FITS related
units can be added by the UnitMap::addFITS() function. For
details, see linkto class="UnitMap">UnitMap.
Example:
The definitions that were current on 990915 are given at end of this file
Explicit use of units is only necessary if:
For these cases a Unit can be defined as either a String or a Unit. If
specified as a Unit an automatic check (with exception if illegal) of
the format of the unit string is performed
See the UnitVal
for details of calculating with units.
See the UnitMap
for the details of defining/viewing named units.
E.g.
The following list of constructors is available.
The following operators and functions are defined on Quantums. They are,
of course, only available if the template Type supports them (e.g. / will
not be defined for a Quantum<String> (whatever that may mean)).
Quanta can be converted to other units by the following set of member
functions:
The value and units of a quantum can be set or retrieved separately by the
following member functions:
The output operator ('<<') will produce the value of the quantum and its
units. Given Quantity myval(5.,"mJy");,
cout << myval; will produce:
"5.0 mJy"; while cout << myval.get("yW/m2") will produce:
".00005 yW/m2.s"
Prerequisite
Etymology
The name Quanta derives from a physical quantity, i.e. a value with
units attached.
Synopsis
The Quanta model deals with units and physical quantities
(i.e. values with a unit).
Units are handled in the Unit section
(see Unit.h).
Quantities are handled in the Quantum section
(see Quantum.h).
In addition the module contains some more general support classes
(Euler angles,
rotation matrix,
pointed string), formatting for
time and angle classes
and classes containing information for
Measures (MeasValue and the derived MV
classes like MVEpoch). See the
MeasValue section.
Includes
Including the casa/Quanta.h will take care of all
includes necessary for the handling of pure Units and Quantities.
Physical units
Physical units are basically used in quantities
(see Quantum), i.e.
a value and a dimension. The Unit class, or one of its subsidiaries, will
in general not be called separately. The only reason to make use of these
classes is to generate additional 'tagged' units, i.e. units with a
special name, e.g. 'beam' for a telescope beam, or 'JY', a non-SI name
for Jy.
Units
A Unit is in principle specified as a String (or directly as "string"),
and can be defined as either a Unit or a String.
If defined as a Unit, the format of the string will be checked for a
legal definition and relevant information (e.g. scale, dimension type) is
cached in the Unit object, leading to (much) faster use; if defined as a
String, the checking will be postponed
until any use is made of the information in the string.
The standard naming conventions for SI units are that they
are all in lowercase, unless derived from a person's name, when they start
with a capital letter. Notable exceptions are some of the astronomical
SI related units (e.g. AU).
A name can be preceded by a (standard) decimal prefix.
km/s/(Mpc.s)2 is identical to km.s-1.Mpc-2.s-2
There are 5 name lists in the UnitMap, which are searched in reverse order:
All known names can be viewed by running the tUnit test program, or
using the MapUnit::list() routine.
There is a difference between units without a dimension (non-dimensioned
I will call them), and undimensioned units. Non-dimensioned examples are
"", "%"; undimensioned examples: "beam", "pixel".
Working with units
In general units are not used explicitly, but are embedded in quantities
and coordinates.
Unit a="km/Ms"; String b="Mm/Gs"; //produce 'identical' units a and b
Unit a("KpH"); // will produce exception
String a("KpH"); // will be accepted till some other action
// done on a
// The following will define a unit named 'tag' with a value identical
// to 5 mJy. After this definition tag can be used as any other unit,
// e.g. Unit("Gtag/pc") will be a valid unit string.
UnitMap::putUser("tag",UnitVal(5.,"mJy"),"my own unit name for 5 mJy");
// The following will calculate how many hp.s per eV
Double hpeV = (UnitVal("hp.s")/UnitVal("eV")).getFac();
// maybe after checking for identical dimensions
if ( UnitVal("hp.s") != UnitVal("eV")) { cout << "unexpected" << endl; }
UnitVal has the following special constants to easily check unit
dimensions (note that they can be combined to e.g. generate velocity
as 'UnitVal::LENGTH/UnitVal::TIME')
Quantums and Quantities
A Quantum is a value with a unit. Quantums are templated on their value
type (e.g. Float, Vector<Double>). Quantity
is a typedef
for the (probably most common) Quantum<Double>.
The basic specification of a Quantum is:
Quantum<Type> ( Type value, Unit unit); // or: String unit or: "unit"
Quantity( Double value, Unit unit); // or: String unit or: "unit"
In the following 'Unit' can be replaced by 'String' (or "string" everywhere.
The only difference being a check for a legitimate unit string being
executed if Unit specified (with exception if error), and a much faster
execution of the Unit is used repeatedly.
Quantum<Type> can, if Type equals Double, be replaced with
Quantity
Quanta can be checked for having the correct unit dimensions (e.g. before
addition or comparing) by the following two member functions, which will
return a Bool value or raise an exception:
QC class of constant quantities
In parallel with the 'C' class of undimensioned constants, the QC class
contains dimensioned constants.
On 960509 the following were defined:
Values for Measures
The MeasValue class derivatives are all named MVmeasure, e.g.
MVFrequency, and represent the internal representation of the
specific measure class. There main use is for the Measures module,
but they can be used alone, e.g. for the conversion to formatted times,
or the conversion of frequencies from say wavelength to frequency.
They all have at least the following constructors:
MV()
MV(MV)
MV(Double)
MV(Vector<Double>)
MV(Quantity)
MV(Vector<Quantity>)
MV(Quantum<Vector<Double> >)
But most have also constructors like:
MV(Double, Double)
MV(Quantity, Quantity)
The actual interpretation is class dependent: see the individual MV classes
like MVEpoch,
MVDirection,
MVPosition,
MVFrequency,
MVDouble,
MVRadialVelocity.
MVBaseline,
MVuvw,
MVEarthMagnetic,
A few examples:
MVEpoch(12345, 0.1e-20) will create one epoch (MJD12345.0), but preserving
the precision of all information
MVDirection(Quantity(20,"deg"), Quantity(-10,"'")) will create a direction
with an RA of 20 degree, and a DEC of -10 arcmin
MVFrequency(Quantity(5,"keV")) will create a frequency corresponding to
the specified energy.
All MVs have the +=, -=, ==, !=, << operators, and near(),
nearAbs(), print() and adjust()
and readjust() (which in general
normalise to a value of 1 (e.g. MVDirection), or recalculates high
precision values (e.g. MVEpoch) functions.
Information can be viewed with many get functions. In most cases
getValue() will return the internal value as either Double or
Vector
In general the Measure classes can be used without worrying about the
MeasValues, since most Measure constructors have enough flexibility (and
their own get()'s) to be able to use them independently).
Special cases are MVAngle and
MVTime, which can do special formatting for
time and angles (in earlier documentation they were called HMS etc.).
Motivation
The Quanta model originated to handle physical quantities independent of their
units.
Units were introduced in the described way to be able to handle any
possible physical unit.
To Do
Example
Known units on 960509
// UnitMap::list() will produce the following list:
List all defined symbols
Prefix table (20):
E (exa) 1e+18
G (giga) 1000000000
M (mega) 1000000
P (peta) 1e+15
T (tera) 1e+12
Y (yotta) 1e+24
Z (zetta) 1e+21
a (atto) 1e-18
c (centi) 0.01
d (deci) 0.1
da (deka) 10
f (femto) 1e-15
h (hecto) 100
k (kilo) 1000
m (milli) 0.001
n (nano) 1e-09
p (pico) 1e-12
u (micro) 1e-06
y (yocto) 1e-24
z (zepto) 1e-21
Defining unit table (10):
A (ampere) 1 A
K (kelvin) 1 K
_ (undimensioned) 1 _
cd (candela) 1 cd
kg (kilogram) 1 kg
m (metre) 1 m
mol (mole) 1 mol
rad (radian) 1 rad
s (second) 1 s
sr (steradian) 1 sr
SI unit table (50):
$ (currency) 1 _
% (percent) 0.01
%% (permille) 0.001
A (ampere) 1 A
AE (astronomical unit) 149597870659 m
AU (astronomical unit) 149597870659 m
Bq (becquerel) 1 s-1
C (coulomb) 1 s.A
F (farad) 1 m-2.kg-1.s4.A2
Gy (gray) 1 m2.s-2
H (henry) 1 m2.kg.s-2.A-2
Hz (hertz) 1 s-1
J (joule) 1 m2.kg.s-2
Jy (jansky) 1e-26 kg.s-2
K (kelvin) 1 K
L (litre) 0.001 m3
M0 (solar mass) 1.98891944407e+30 kg
N (newton) 1 m.kg.s-2
Ohm (ohm) 1 m2.kg.s-3.A-2
Pa (pascal) 1 m-1.kg.s-2
S (siemens) 1 m-2.kg-1.s3.A2
S0 (solar mass) 1.98891944407e+30 kg
Sv (sievert) 1 m2.s-2
T (tesla) 1 kg.s-2.A-1
UA (astronomical unit) 149597870659 m
V (volt) 1 m2.kg.s-3.A-1
W (watt) 1 m2.kg.s-3
Wb (weber) 1 m2.kg.s-2.A-1
_ (undimensioned) 1 _
a (year) 31557600 s
arcmin (arcmin) 0.000290888208666 rad
arcsec (arcsec) 4.8481368111e-06 rad
as (arcsec) 4.8481368111e-06 rad
cd (candela) 1 cd
cy (century) 3155760000 s
d (day) 86400 s
deg (degree) 0.0174532925199 rad
g (gram) 0.001 kg
h (hour) 3600 s
l (litre) 0.001 m3
lm (lumen) 1 cd.sr
lx (lux) 1 m-2.cd.sr
m (metre) 1 m
min (minute) 60 s
mol (mole) 1 mol
pc (parsec) 3.08567758065e+16 m
rad (radian) 1 rad
s (second) 1 s
sr (steradian) 1 sr
t (tonne) 1000 kg
Customary unit table (74):
" (arcsec) 4.8481368111e-06 rad
"_2 (square arcsec) 2.35044305391e-11 sr
' (arcmin) 0.000290888208666 rad
'' (arcsec) 4.8481368111e-06 rad
''_2 (square arcsec) 2.35044305391e-11 sr
'_2 (square arcmin) 8.46159499408e-08 sr
: (hour) 3600 s
:: (minute) 60 s
::: (second) 1 s
Ah (ampere hour) 3600 s.A
Angstrom (angstrom) 1e-10 m
Btu (British thermal unit (Int)) 1055.056 m2.kg.s-2
CM (metric carat) 0.0002 kg
Cal (large calorie (Int)) 4186.8 m2.kg.s-2
FU (flux unit) 1e-26 kg.s-2
G (gauss) 0.0001 kg.s-2.A-1
Gal (gal) 0.01 m.s-2
Gb (gilbert) 0.795774715459 A
Mx (maxwell) 1e-08 m2.kg.s-2.A-1
Oe (oersted) 79.5774715459 m-1.A
R (mile) 0.000258 kg-1.s.A
St (stokes) 0.0001 m2.s-1
Torr (torr) 133.322368421 m-1.kg.s-2
USfl_oz (fluid ounce (US)) 2.95735295625e-05 m3
USgal (gallon (US)) 0.003785411784 m3
WU (WSRT flux unit) 5e-29 kg.s-2
abA (abampere) 10 A
abC (abcoulomb) 10 s.A
abF (abfarad) 1000000000 m-2.kg-1.s4.A2
abH (abhenry) 1e-09 m2.kg.s-2.A-2
abOhm (abohm) 1e-09 m2.kg.s-3.A-2
abV (abvolt) 1e-08 m2.kg.s-3.A-1
ac (acre) 4046.8564224 m2
arcmin_2 (square arcmin) 8.46159499408e-08 sr
arcsec_2 (square arcsec) 2.35044305391e-11 sr
ata (technical atmosphere) 98066.5 m-1.kg.s-2
atm (standard atmosphere) 101325 m-1.kg.s-2
bar (bar) 100000 m-1.kg.s-2
beam (undefined beam area) 1 _
cal (calorie (Int)) 4.1868 m2.kg.s-2
cwt (hundredweight) 50.80234544 kg
deg_2 (square degree) 0.000304617419787 sr
dyn (dyne) 1e-05 m.kg.s-2
eV (electron volt) 1.60217733e-19 m2.kg.s-2
erg (erg) 1e-07 m2.kg.s-2
fl_oz (fluid ounce (Imp)) 2.84130488996e-05 m3
ft (foot) 0.3048 m
fu (flux unit) 1e-26 kg.s-2
fur (furlong) 201.168 m
gal (gallon (Imp)) 0.00454608782394 m3
ha (hectare) 10000 m2
hp (horsepower) 745.7 m2.kg.s-3
in (inch) 0.0254 m
kn (knot (Imp)) 0.514773333333 m.s-1
lb (pound (avoirdupois)) 0.45359237 kg
ly (light year) 9.46073047e+15 m
mHg (metre of mercury) 133322.387415 m-1.kg.s-2
mile (mile) 1609.344 m
n_mile (nautical mile (Imp)) 1853.184 m
oz (ounce (avoirdupois)) 0.028349523125 kg
pixel (pixel) 1 _
sb (stilb) 10000 m-2.cd
sq_arcmin (square arcmin) 8.46159499408e-08 sr
sq_arcsec (square arcsec) 2.35044305391e-11 sr
sq_deg (square degree) 0.000304617419787 sr
statA (statampere) 3.33564095198e-10 A
statC (statcoulomb) 3.33564095198e-10 s.A
statF (statfarad) 1.11188031733e-12 m-2.kg-1.s4.A2
statH (stathenry) 899377374000 m2.kg.s-2.A-2
statOhm (statohm) 899377374000 m2.kg.s-3.A-2
statV (statvolt) 299.792458 m2.kg.s-3.A-1
u (atomic mass unit) 1.661e-27 kg
yd (yard) 0.9144 m
yr (year) 31557600 s
Classes