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Observations: Photometry, Spectroscopy, and Astrometry
In order to determine the coordinates of a star in 7D phase space, a variety of astronomical techniques must be used. As always, the most crucial quantity to measure is stellar distance. Photometric parallax method are always used to estimate stellar distance, and therefor accurate Photometry is impotrant. For certain populations, for example RR Lyrae stars, a good estimate of absolute magnitude is obtained as a simple constant (with some metallicity dependence); for other populations,such as main-sequence stars, the absolute magnitude depends on both effective temperature and metallicity, and sometimes on age (or surface gravity) as well. Accurate measurements of stellar metallicity acquires Spectroscopic observations. To measure all three components of the space-velocity vector, precise Astrometric observations are also required. Therefore, multicolor imaging, multiepoch astrometry, and spectroscopy are required for measuring the coordinates of a star in the 7D position-velocity-metallicity phase space.

Stellar metallicity,together with effective temperature and surface gravity, is one of the three main parameters that affect the observed spectral energy distribution of most stars.

Metallicity for the MW stars
Metallicity & alpha-enhancement(Salaris & Cassisi, Evolution of Stars and Stellar Populations)

The traditional metal abundance indicator is the quantity that the difference of the logarithm of the Fe/H number abundance ratios observed in the atmosphere of the target star and in the solar one:

$$[Fe/H]=log(N(Fe)/N(H))_*-log(N(Fe)/N(H))_{sun}$$

The choice of iron as the metal abundance indicator (in spite of the fact that in the Sun iron is not one of the most abundant metals) stems from the fact that iron lines are prominent and easy to measure. If one assumes that the solar heavy elment distribution is universal, the conversion from Z to [Fe/H] is given by

$$[Fe/H]=log(Z/X)_*-log(Z/X)_{sun}$$

because $$X_{Fe}$$ is the same in the Sun and in the target star. When cccounting for the solar value of (Z/X) it becomes

$$[Fe/H]=log(Z/X)_*+1.61=log(Z/(1-Y-Z))_*+1.61$$

Generally, the helium and heavy element abundance variations are negligible with respect to he hydrogen abundance that largely dominates the stellar atmosspheres apart from specific cases. This means that X can be considered approximately a constant and previous eq. can be simplified into

$$[Fe/H] = log(Z_*/Z_{sun})$$

If one relaxes the assumption of a universal scaled solar heavy element distribution,the correspondence between [Fe/H] and X,Y,Z obviously changes because the ratio between the iron abundance and Z is different from that in the Sun, and depends on the exact metal distribution in the observed star. In this case,

$$[M/H]=[Fe/H]=log(Z/X)_*-log(Z/X)_{sun}$$

A very important case is the alpha-enhanced metal distribution typical of old metal-poor([Fe/H] below about -0.6) objects. O,Ne,Mg,Si,S,Ca and Ti is the so-called alpha elemnts). For these alpha-enhanced mixtures the general relationship between [M/H] and [Fe/H] is well approximated by

$$[M/H] = [Fe/H]+log(0.694*f_{\alpha} + 0.306)$$

where $$f_{\alpha}=10^{[\alpha/Fe]}$$. It is easy to understand why with this specific non-scaled solar metal mixture [M/H] > [Fe/H]. The ratio $$\alpha/Fe$$ is larger than the solar counter part at given Fe abundance, and therefore the same Fe abundance corresponds to a total metal abundance larger than the case of a scaled solar one.

Age-metallicity relation (AMR)(Rocha-Pinto et al. 2000, A&A,358,850)

Star Formation

Photometric Parallax and Photometric metallicity

In this paper of Juric et al, 2008,ApJ,673,864,photometric parallax is estimated by

bright normalization: $$M_r=3.2+13.30*(r-i)-11.50*(r-i)^2+5.40*(r-i)^3-0.70*(r-i)^4$$

faint normalization: $$M_r=4.0+11.86*(r-i)-10.74*(r-i)^2+5.99*(r-i)^3-1.20*(r-i)^4$$

Improved photometric parallax is expressed as follows by Ivezic et al 2008,ApJ,684,287, combination of equations (A2) and (A7) in their Appendix A

$$M_r=-0.56+14.32*x-12.97*x^2+6.127*x^3-1.267*x^4+0.0967*x^5-1.11*y-0.18*y^2 $$

where x=g-i,y=[Fe/H]

for stars of g-r < 0.6 & g-r > 0.2, the metallicity [Fe/H] expressed by Bond 2010,ApJ,716,1:

$$[Fe/H]=A+Bx+Cy+Dxy+Ex^2+Fy^2+Gx^2y+Hxy^2+Ix^3+Jy^3$$

where x=u-g and y=g-r, the best-fit coefficients (A to J):−13.13, 14.09, 28.04, −5.51,−5.90, −58.68, 9.14, −20.61, 0.0, 58.20.

A simple Model for the MW's Metallicity distribution(Ivezic,2008,ApJ,684,287):

$$prob([Fe/H])=(1-f_H)G(\mu_D,\sigma_D)+f_HG(\mu_H,\sigma_H)$$

$$G(\mu,\sigma)=e^{-(x-\mu)^2/2\sigma^2}/\sqrt{2\pi}\sigma$$

for halo stars:$$\mu_H=-1.46, \sigma_H=0.30, f_H=1/(1+aexp[-(|Z|/b)^c]),(a,b,c)=(70,240pc,0.62)$$,

for thick-disk stars:$$\sigma_{D2}=0.21, \mu_{D2} $$ is a function of height:

$$\mu_D=\mu_0+\Delta_{\mu}exp(-|Z|/H_{\mu})$$ dex

where $$H_{\mu}=1.0kpc,\mu_0=-0.78,\Delta_{\mu}=0.35$$

for thin-disk stars:$$\sigma_{D1}=0.11, \mu_{D1}=\mu_{D2}+0.14 $$

Milky Way kinematic
Some basic defines: proper motion



where δ is the declination. The factor in cos δ accounts for the fact that the radius from the axis of the sphere to its surface varies as cos δ. Tangential velocity, (VT), of a star is the across the line-of-sight velocity expressed in units of kms-1. VT can be derived from the proper motion (mu) and the distance (d) in parsecs of the star via
 * style="text-align: right;"|$$\ \mu^2$$
 * $$={\mu_\delta}^2 + {\mu_\alpha}^2 \cdot \cos^2 \delta \ ,$$
 * }
 * }

$$ V_T [{kms}^{-1}]  =  4.74  \mu [arcsec {yr}^{-1}] d [pc] $$

You should confirm the unit conversion factor of 4.74. ( 1 radian = 206265 arcsecond, 1 parsec = 206265 AU, 1 AU = 149,600,000 km ) The 3D Space Motion, (VS), is derived from VR and VT, via,

$$ {V_S}^2 = {V_R}^2 + {V_T}^2 $$

Unerstand the defination of $$ V_R,V_{\phi},V_Z $$ in the Galactocentric coordinate system. Please read the following paper for understanding MW's kinematics:[http://adsabs.harvard.edu/abs/2010ApJ...716....1B '''THE MILKY WAY TOMOGRAPHY WITH SDSS. III. STELLAR KINEMATICS''']

Binary system(review article: Stellar Multiplicity,ARAA,2013,51,269)
Since about half of the stars are known to be in multiple systems,the HST LF misses essentially all companions.