# Magnitudes and other Logarithmic Units¶

Magnitudes and logarithmic units such as dex and dB are used the logarithm of values relative to some reference value. Quantities with such units are supported in astropy via the Magnitude, Dex, and Decibel classes.

## Creating Logarithmic Quantities¶

One can create logarithmic quantities either directly or by multiplication with a logarithmic unit. For instance:

>>> import astropy.units as u, astropy.constants as c, numpy as np
>>> u.Magnitude(-10.)
<Magnitude -10. mag>
>>> u.Magnitude(10 * u.ct / u.s)
<Magnitude -2.5 mag(ct / s)>
>>> u.Magnitude(-2.5, "mag(ct/s)")
<Magnitude -2.5 mag(ct / s)>
>>> -2.5 * u.mag(u.ct / u.s)
<Magnitude -2.5 mag(ct / s)>
>>> u.Dex((c.G * u.M_sun / u.R_sun**2).cgs)
<Dex 4.438067627303133 dex(cm / s2)>
>>> np.linspace(2., 5., 7) * u.Unit("dex(cm/s2)")
<Dex [2. , 2.5, 3. , 3.5, 4. , 4.5, 5. ] dex(cm / s2)>


Above, we make use of the fact that the units mag, dex, and dB are special in that, when used as functions, they return a LogUnit instance (MagUnit, DexUnit, and DecibelUnit, respectively). The same happens as required when strings are parsed by Unit.

As for normal Quantity objects, one can access the value with the value attribute. In addition, one can convert easily to a Quantity with the physical unit using the physical attribute:

>>> logg = 5. * u.dex(u.cm / u.s**2)
>>> logg.value
5.0
>>> logg.physical
<Quantity 100000. cm / s2>


## Converting to different units¶

Like Quantity objects, logarithmic quantities can be converted to different units, be it another logarithmic unit or a physical one:

>>> logg = 5. * u.dex(u.cm / u.s**2)
>>> logg.to(u.m / u.s**2)
<Quantity 1000. m / s2>
>>> logg.to('dex(m/s2)')
<Dex 3. dex(m / s2)>


For convenience, the si and cgs attributes can be used to convert the Quantity to base S.I. or c.g.s units:

>>> logg.si
<Dex 3. dex(m / s2)>


## Arithmetic and Photometric Applications¶

Addition and subtraction work as expected for logarithmic quantities, multiplying and dividing the physical units as appropriate. It may be best seen through an example of a very simple photometric reduction. First, calculate instrumental magnitudes assuming some count rates for three objects:

>>> tint = 1000.*u.s
>>> cr_b = ([3000., 100., 15.] * u.ct) / tint
>>> cr_v = ([4000., 90., 25.] * u.ct) / tint
>>> b_i, v_i = u.Magnitude(cr_b), u.Magnitude(cr_v)
>>> b_i, v_i
(<Magnitude [-1.19280314,  2.5       ,  4.55977185] mag(ct / s)>,
<Magnitude [-1.50514998,  2.61439373,  4.00514998] mag(ct / s)>)


Then, the instrumental B-V color is simply:

>>> b_i - v_i
<Magnitude [ 0.31234684, -0.11439373,  0.55462187] mag>


Note that the physical unit has become dimensionless. The following step might be used to correct for atmospheric extinction:

>>> atm_ext_b, atm_ext_v = 0.12 * u.mag, 0.08 * u.mag
>>> secz = 1./np.cos(45 * u.deg)
>>> b_i0 = b_i - atm_ext_b * secz
>>> v_i0 = v_i - atm_ext_b * secz
>>> b_i0, v_i0
(<Magnitude [-1.36250876,  2.33029437,  4.39006622] mag(ct / s)>,
<Magnitude [-1.67485561,  2.4446881 ,  3.83544435] mag(ct / s)>)


Since the extinction is dimensionless, the units do not change. Now suppose the first star has a known ST magnitude, so we can calculate zero points:

>>> b_ref, v_ref = 17.2 * u.STmag, 17.0 * u.STmag
>>> b_ref, v_ref
(<Magnitude 17.2 mag(ST)>, <Magnitude 17. mag(ST)>)
>>> zp_b, zp_v = b_ref - b_i0[0], v_ref - v_i0[0]
>>> zp_b, zp_v
(<Magnitude 18.56250876 mag(s ST / ct)>,
<Magnitude 18.67485561 mag(s ST / ct)>)


Here, ST is a short-hand for the ST zero-point flux:

>>> (0. * u.STmag).to(u.erg/u.s/u.cm**2/u.AA)
<Quantity 3.63078055e-09 erg / (Angstrom cm2 s)>
>>> (-21.1 * u.STmag).to(u.erg/u.s/u.cm**2/u.AA)
<Quantity 1. erg / (Angstrom cm2 s)>


Note

at present, only magnitudes defined in terms of luminosity or flux are implemented, since those that do not depend on the filter the measurement was made with. They include absolute and apparent bolometric [M15], ST [H95] and AB [OG83] magnitudes.

Now applying the calibration, we find (note the proper change in units):

>>> B, V = b_i0 + zp_b, v_i0 + zp_v
>>> B, V
(<Magnitude [17.2       , 20.89280314, 22.95257499] mag(ST)>,
<Magnitude [17.        , 21.1195437 , 22.51029996] mag(ST)>)


We could convert these magnitudes to another system, e.g., ABMag, using appropriate equivalency:

>>> V.to(u.ABmag, u.spectral_density(5500.*u.AA))
<Magnitude [16.99023831, 21.10978201, 22.50053827] mag(AB)>


Suppose we also knew the intrinsic color of the first star, then we can calculate the reddening:

>>> B_V0 = -0.2 * u.mag
>>> EB_V = (B - V)[0] - B_V0
>>> R_V = 3.1
>>> A_V = R_V * EB_V
>>> A_B = (R_V+1) * EB_V
>>> EB_V, A_V, A_B
(<Magnitude 0.4 mag>, <Quantity 1.24 mag>, <Quantity 1.64 mag>)


Here, one sees that the extinctions have been converted to quantities. This happens generally for division and multiplication, since these processes work only for dimensionless magnitudes (otherwise, the physical unit would have to be raised to some power), and Quantity objects, unlike logarithmic quantities, allow units like mag / d.

Note that one can take the automatic unit conversion quite far (perhaps too far, but it is fun). For instance, suppose we also knew the bolometric correction and absolute bolometric magnitude, then we can calculate the distance modulus:

>>> BC_V = -0.3 * (u.m_bol - u.STmag)
>>> M_bol = 5.46 * u.M_bol
>>> DM = V[0] - A_V + BC_V - M_bol
>>> BC_V, M_bol, DM
(<Magnitude -0.3 mag(bol / ST)>,
<Magnitude 5.46 mag(Bol)>,
<Magnitude 10. mag(bol / Bol)>)


With a proper equivalency, we can also convert to distance without remembering the 5-5log rule:

>>> radius_and_inverse_area = [(u.pc, u.pc**-2,
...                            lambda x: 1./(4.*np.pi*x**2),
...                            lambda x: np.sqrt(1./(4.*np.pi*x)))]
<Quantity 1000. pc>


## Numpy functions¶

For logarithmic quantities, most numpy functions and many array methods do not make sense, hence they are disabled. But one can use those one would expect to work:

>>> np.max(v_i)
<Magnitude 4.00514998 mag(ct / s)>
>>> np.std(v_i)
<Magnitude 2.33971149 mag>


Note

This is implemented by having a list of supported ufuncs in units/function/core.py and by explicitly disabling some array methods in FunctionQuantity. If you believe a function or method is incorrectly treated, please let us know.

## Dimensionless logarithmic quantities¶

Dimensionless quantities are treated somewhat specially, in that, if needed, logarithmic quantities will be converted to normal Quantity objects with the appropriate unit of mag, dB, or dex. With this, it is possible to use composite units like mag/d or dB/m, which cannot easily be supported as logarithmic units. For instance:

>>> dBm = u.dB(u.mW)
>>> signal_in, signal_out = 100. * dBm, 50 * dBm
>>> cable_loss = (signal_in - signal_out) / (100. * u.m)
>>> signal_in, signal_out, cable_loss
(<Decibel 100. dB(mW)>, <Decibel 50. dB(mW)>, <Quantity 0.5 dB / m>)
>>> better_cable_loss = 0.2 * u.dB / u.m
>>> signal_in - better_cable_loss * 100. * u.m
<Decibel 80. dB(mW)>

 [M15] Mamajek et al., 2015, arXiv:1510.06262
 [H95] E.g., Holtzman et al., 1995, PASP 107, 1065
 [OG83] Oke, J.B., & Gunn, J. E., 1983, ApJ 266, 713