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- Magnitudes and Other Logarithmic Units

# Magnitudes and Other Logarithmic Units

Magnitudes and logarithmic units such as `dex` and `dB` are used as 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

You can create logarithmic quantities either directly or by multiplication with a logarithmic unit.

### Example

To create a logarithmic quantity:

    >>> 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, you can access the value with the `value` attribute. In addition, you can convert 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.

### Example

To convert a logarithmic quantity to a different unit:

    >>> 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 SI or CGS 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 photometric reduction.

### Example

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:

    >>> 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 shorthand 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 do not depend on the filter with which the measurement was made. 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, for example, ABMag, using appropriate equivalency:

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

This is particularly useful for converting magnitude into flux density. `V` is currently in ST magnitudes, which is based on flux densities per unit wavelength (\(f_\lambda\)). Therefore, we can directly convert `V` into flux density per unit wavelength using the `to()` method:

    >>> flam = V.to(u.erg/u.s/u.cm**2/u.AA)
    >>> flam
    <Quantity [5.75439937e-16, 1.29473986e-17, 3.59649961e-18] erg / (Angstrom cm2 s)>

To convert `V` to flux density per unit frequency (\(f_\nu\)), we again need the appropriate equivalency, which in this case is the central wavelength of the magnitude band, 5500 Angstroms:

    >>> lam = 5500 * u.AA
    >>> fnu = V.to(u.erg/u.s/u.cm**2/u.Hz, u.spectral_density(lam))
    >>> fnu
    <Quantity [5.80636959e-27, 1.30643316e-28, 3.62898099e-29] erg / (cm2 Hz s)>

We could have used the central frequency instead:

    >>> nu = 5.45077196e+14 * u.Hz
    >>> fnu = V.to(u.erg/u.s/u.cm**2/u.Hz, u.spectral_density(nu))
    >>> fnu
    <Quantity [5.80636959e-27, 1.30643316e-28, 3.62898099e-29] erg / (cm2 Hz s)>

Note

When converting magnitudes to flux densities, the order of operations matters; the value of the unit needs to be established before the conversion. For example, `21 * u.ABmag.to(u.erg/u.s/u.cm**2/u.Hz)` will give you 21 times \(f_\nu\) for an AB mag of 1, whereas `(21 * u.ABmag).to(u.erg/u.s/u.cm**2/u.Hz)` will give you \(f_\nu\) for an AB mag of 21.

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>, <Magnitude 1.24 mag>, <Magnitude 1.64 mag>)

Here, you see 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 you 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 (but you might find the `Distance` class to be even more convenient):

    >>> 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)))]
    >>> DM.to(u.pc, equivalencies=radius_and_inverse_area)
    <Quantity 1000. pc>

## NumPy Functions

For logarithmic quantities, most `numpy` functions and many array methods do not make sense, hence they are disabled. But you can use those you 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 conveniently 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)>

References

[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