The Astrobiological Copernican Weak and Strong Limits for Intelligent Life

The first parameter we investigate is f_L—the fraction of stars within the the Milky Way that are older than 5 Gyr. This relates to our assumption that communicative intelligent life can form after this time, which we are making based on the fact that intelligent life on the Earth took approximately this long to develop (see Dalrymple 2001; note: we have taken the best estimate for the age of the Earth as ${4.54}_{-0.05}^{+0.05}\,\mathrm{Gyr}$, and we express this to one significant figure). Clearly, this is the only timescale we have as an example for the formation of intelligent life, and it is, perforce, a simplified first approximation: we are considering the persistence of the stability of the star's conditions, and can say nothing about the planetary environment, which may be dramatically affected by climate, orbital, or geological shifts within this 5 Gyr timeframe. In fact, because of this our limits are, in many ways, upper limits due to these other conditions. However, even if we relax this assumption, we find very little difference in the following results (see Section 3.1.4), and we demonstrate that, even within this apparently severe limit, we obtain interesting results.

To carry out this calculation we need to determine the age distribution of stars within our Galaxy. To do this, we assume a form of the SFR history and then convert this into the number of stars formed at a given mass as a function of time, throughout the entire history of the Galaxy. To do this first part, we use an analytical fit for data on SFR versus redshift (z).

What we want to use here is a function that describes the variation in SFR within the the Milky Way throughout time; however, this is difficult to know and no functional formula or derivation of this exists. We start by using established SFR data for distant galaxies throughout the universe at large, as reported by Madau & Dickinson (2014): the data from Table 1 of that paper, showing log(SFR/M_⊙ yr⁻¹ Mpc⁻³), together with their associated error bars, versus redshift, z, is used in the construction of Figure 1. This is likely a good presentation of past SFR within the Milky Way as the star formation history of the Local Group matches the global history fairly well (e.g., Weisz et al. 2014). We do not have the exact values of the Milky Way's star formation history, so we use knowledge of the shape of the global history and renormalize this based on the known volume of our Galaxy and the number of stars it contains today.

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To do this we take the analytical form of the SFR history of the universe and adapt it as the relative star formation history of the Milky Way, which is the relevant quantity for this work. For this we use the variation of SFR with redshift which can be fitted as an analytical expression, as shown by many authors (e.g., Hernquist & Springel 2003; Hopkins 2004; Hopkins & Beacom 2006). In this last work, observed data on SFR at different redshifts, z, (representing different times in cosmic history) are fitted with a function of the following form:

$$\begin{eqnarray}&&\dot{{\rho }_{* }}\left(z\right)=\ \dot{{\rho }_{* }}\left(0\right)\ \displaystyle \frac{{\chi }^{{n}_{1}}}{1+\ \alpha {\left(\chi -1\right)}^{{n}_{2}}\exp (\beta {\chi }^{{n}_{3}})}\end{eqnarray} \tag{ 4 }$$

where:

$\dot{{\rho }_{* }}\left(z\right)$ = SFR, in units M_⊙ Mpc⁻³ yr⁻¹, as a function of redshift,

$\dot{{\rho }_{* }}\left(0\right)$ = present-day value of SFR

and χ is a function of redshift, $\left(z\right)$, which is defined by

$$\begin{eqnarray*}&&\chi \ \left(z\right)={\left(\displaystyle \frac{H(z)}{{H}_{0}}\right)}^{\tfrac{2}{3}}.\end{eqnarray*}$$

H(z) represents the Hubble parameter as a function of redshift, in units km s⁻¹ Mpc⁻¹, and is defined by

$$\begin{eqnarray*}&&H\left(z\right)={H}_{0}{\left[\left({{\rm{\Omega }}}_{M}{\left(1+z\right)}^{3}+\left(1-{{\rm{\Omega }}}_{M}-{{\rm{\Omega }}}_{{\rm{\Lambda }}}\right){\left(1+z\right)}^{2}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}\right)\right]}^{\tfrac{1}{2}}.\end{eqnarray*}$$

Here, H₀ is the present-day value of the Hubble constant, taken as 70 km s⁻¹ Mpc⁻¹, and Ω_M (the density parameter of matter), and Ω_Λ (the density parameter of dark energy) are taken as 0.3 and 0.7 respectively in this work.

α, β, n₁, n₂, and n₃ are constants whose values may be varied to achieve the best fit to the observed data for SFR versus z. Curve-fitting techniques yield the values of the parameters that give a best fit to the observational data. This fit yields the following values for the fitting constants in Equation (4), shown in Table 2.

Table 2.  Fitting Constants for Equation (4)

Fitting	Value for Fitting Constant (with	Range of Values of fL Obtained from
Constant	Associated Range, Where Possible)	Figure 3, Based on the Range in this
	Derived from Curve-fitting Analysis	Fitting Constant
α	${0.528}_{-0.438}^{+1.472}$	0.967 < fL < 0.973
β	${2.36}_{-0.96}^{+0.64}$	0.926 < fL < 0.996
$\dot{{\rho }_{* }}$(0)	${0.330}_{-0.230}^{+1.670}$	0.968 < fL < 0.968
n1	${5.91}_{-1.91}^{+1.47}$	0.906 < fL < 0.997
n2	−0.3508 (no range; see below)	0.968 (no range; see below)
n3	${1.22}_{-0.42}^{+0.68}$	0.782 < fL < 0.999

Note. The curve-fitting analysis was highly volatile to slight changes in the parameter n₂, so this was not varied to allow for an exploration of the range of the other fitting constants.

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We then combine this relationship of the SFR with redshift with that between redshift and lookback time, t_L (i.e., the time between the emission of the light from a distant source, and the present time at which the light is received by the observer). We then plot the cosmic history of star formation (see Figure 1), in which the raw SFR data have been combined with the analytical fitting function, Equation (4), using parameters from Table 2. Note that the raw data have associated error bars, and we have shifted some points slightly to avoid overlap. Note also that in Figure 1 the green curve shows the fitting function in which the central values of the fitting constants are employed.

(i) Distribution of stellar masses according to the Salpeter IMF

At this point, we must introduce the IMF, in order to calculate the distribution of the numbers of stars, N, with masses, M, between certain mass limits (M_lower and M_upper). The starting point we consider here is the Salpeter IMF (Salpeter 1955), described in the following way:

$$\begin{eqnarray}&&\tfrac{{dN}}{{dM}}={{kM}}^{\alpha }\end{eqnarray} \tag{ 5 }$$

where k is a normalization constant, and α = −2.35. Hence

$$\begin{eqnarray*}&&\int {dN}=k.{\int }_{{M}_{\mathrm{lower}}}^{{M}_{\mathrm{upper}}}{M}^{\alpha }{dM}\end{eqnarray*}$$

and therefore

$$\begin{eqnarray}&&N=\tfrac{k}{\alpha +1}.{\left[{M}^{\alpha +1}\right]}_{{M}_{\mathrm{lower}}}^{{M}_{\mathrm{upper}}}.\end{eqnarray} \tag{ 6 }$$

If we normalize this expression such that N = 1, this allows us to find the relative fraction of stars in the entire stellar population which formed with masses between any desired limits. To do this, we evaluate the normalization constant k using values for the minimum and maximum possible stellar mass as M_lower and M_upper respectively.

The minimum mass required for hydrogen fusion is taken as 0.08 M_⊙ (Richer et al. 2006; note: this choice of lower mass limit may have an impact on the final value for f_L; see Section 3.1.4), while the maximum stellar mass we use is 100 M_⊙ (Kroupa 2005); there is considerable debate on this upper limit, but fortunately this value will prove far less significant at the higher end of the Salpeter distribution.

Using these values, we set the left-hand side of Equation (6) equal to unity, in order to achieve an expression for the relative fraction of all stars between the given mass limits. Therefore, we evaluate k = 0.0446. Hence, we can rewrite Equation (6):

$$\begin{eqnarray}&&\tfrac{N}{{N}_{\mathrm{total}}}=0.033\ \times {\left[{M}^{\alpha +1}\right]}_{{M}_{\mathrm{upper}}}^{{M}_{\mathrm{lower}}}\end{eqnarray} \tag{ 7 }$$

which we use to evaluate the relative fraction of stars in a population with masses between the desired limits. We explore the relationship between this survival fraction and the time since starburst and find that the vast majority of the stars that still survive today did indeed form in starbursts more than 5 Gyr ago.

(ii) Stellar masses and lifetimes

We now consider the main-sequence lifetime of stars, in order to ascertain the fraction of stars that formed at a certain time in the past and still survive today. The first step here is to establish the relationship between stellar mass and a star's lifetime on the main sequence. The amount of time that a star spends burning hydrogen will, of course, depend on its initial mass and luminosity, as: ${\tau }_{\mathrm{MS}}\ \sim \ \left(\tfrac{M}{L}\right)$. Given that the estimated main-sequence lifetime of the Sun is of order 10 Gyr (Schroder & Smith 2008), we may write, in terms of solar units,

$$\begin{eqnarray}&&\tfrac{{\tau }_{\mathrm{MS}}}{\mathrm{Gyr}}\ \sim \ 10\left(\tfrac{M}{{M}_{\odot }}\right){\left(\tfrac{L}{{L}_{\odot }}\right)}^{-1}\end{eqnarray} \tag{ 8 }$$

where L is the luminosity of the star. Luminosity is also tightly dependent on stellar mass and, taking the relationships from Salaris & Cassisi (2005),

$$\begin{eqnarray}\begin{array}{rcl}\tfrac{{L}_{\mathrm{MS}}}{{L}_{\odot }} & \approx & \ 1.4{\left(\tfrac{M}{{M}_{\odot }}\right)}^{3.5}\ \mathrm{for}\ 2\,{M}_{\odot }\leqslant M\leqslant 20\,{M}_{\odot }\\ \tfrac{{L}_{\mathrm{MS}}}{{L}_{\odot }} & \approx & \ {\left(\tfrac{M}{{M}_{\odot }}\right)}^{4}\ \mathrm{for}\ 0.43\,{M}_{\odot }\leqslant M\lt 2\,{M}_{\odot }\\ \tfrac{{L}_{\mathrm{MS}}}{{L}_{\odot }} & \approx & \ 0.23{\left(\tfrac{M}{{M}_{\odot }}\right)}^{2.3}\ \mathrm{for}\ M\lt 0.43\,{M}_{\odot }.\end{array}\end{eqnarray} \tag{ 9 }$$

Hence, combining the relationships from Equations (8) and (9) we find

$$\begin{eqnarray}\begin{array}{rcl}{\tau }_{\mathrm{MS}}/\mathrm{Gyr} & \approx & \ 7.1{\left(\tfrac{M}{{M}_{\odot }}\right)}^{-2.5}\ \mathrm{for}\ 2\,{M}_{\odot }\leqslant M\leqslant 20\,{M}_{\odot }\\ {\tau }_{\mathrm{MS}}/\mathrm{Gyr} & \approx & \ 10{\left(\tfrac{M}{{M}_{\odot }}\right)}^{-3}\ \mathrm{for}\ 0.43\,{M}_{\odot }\leqslant M\lt 2\,{M}_{\odot }\\ {\tau }_{\mathrm{MS}}/\mathrm{Gyr} & \approx & \ 43{\left(\tfrac{M}{{M}_{\odot }}\right)}^{-1.3}\ \mathrm{for}\ M\lt 0.43\,{M}_{\odot }.\end{array}\end{eqnarray} \tag{ 10 }$$

Since we are interested in evaluating the fraction of all stars that are older than 5 Gyr, we examine only stars of spectral types F, G, K, M, and lower (if we include L- and T-type dwarfs in our consideration), for which we take the mass data in Table 3.

For example, a stellar mass value of 1.26 M_⊙ corresponds to a main-sequence lifetime of 5 Gyr, which is the critical value we use in our determination of f_L.

One conclusion from our results is that the most likely location for CETI life is around low-mass M-dwarf stars, since we estimate that these contribute around 90% of the total population. It is thus important to note that one major issue in astrobiology and the development of life outside the Earth is the lack of understanding as to whether M-dwarf stars are likely hosts for the stable conditions required for the development of intelligent, or even basic, life. This is a particular problem for the present analysis, since the numbers of low-mass, long-lived stars will be dominated by these M dwarfs. This is an issue of ongoing debate, and new exoplanet discoveries in M-dwarf systems (such as Proxima Centauri and Trappist-1) raise new questions about the planetary environments, especially for small planets in close, tidally-locked orbits around volatile, small stars. Wandel (2018) presents a contemporary discussion on these issues, and Haqq-Misra et al. (2018) consider the implications of our existence around a (less abundant) yellow star, as opposed to a (more abundant) red star, in light of the Copernican Principle. It is hoped that future exoplanet discoveries will help to refine the relative abundance of stable planetary environments around M-dwarf stars.

In this section we investigate the survival fraction of stars in the Milky Way over time and use this to arrive at the age distribution of all stars surviving in the Galaxy today. First, we use the analytical fit to the raw SFR data to calculate the total mass of stars formed per Mpc³, per 50 Myr interval of time, as a function of the time since each star-forming event.

However, more significant for our purpose is the mass of stars per Mpc³ that was formed during this 50 Myr time-step, and still survives today. We then renormalize this such that the integral of the number of these stars is equal to the total number we know exist today in the Milky Way. To achieve this, we use the Salpeter IMF to calculate the distribution of the total mass of stars made up by stars of each mass. The number of stars formed in a particular mass category, N, is given by Equation (6). Therefore, the total mass of the stars within each mass category is the number of stars of a given mass multiplied by that mass, giving

$$\begin{eqnarray}\begin{array}{rcl}{M}_{\mathrm{total}} & = & b.{\displaystyle \int }_{{M}_{\mathrm{lower}}}^{{M}_{\mathrm{upper}}}M.{M}^{\alpha }.{dM}\\ {M}_{\mathrm{total}} & = & \displaystyle \frac{b}{\alpha +2}.{\left[{M}^{\alpha +2}\right]}_{{M}_{\mathrm{lower}}}^{{M}_{\mathrm{upper}}}\end{array}\end{eqnarray} \tag{ 11 }$$

where b is another normalization constant, and again, α = −2.35 for a Salpeter IMF.

We normalize this expression to make M_total = 1 for the full range of masses, from M_lower = 0.08 M_⊙ to M_upper = 100 M_⊙, giving

$$\begin{eqnarray}&&\displaystyle \frac{M}{{M}_{\mathrm{total}}}=0.450\ \times {\left[{M}^{-0.35}\right]}_{{M}_{\mathrm{upper}}}^{{M}_{\mathrm{lower}}}\end{eqnarray} \tag{ 12 }$$

which can be used to find the relative fraction of the grand total of stellar mass that was formed in a particular category of stars of masses between the desired limits. (Note: once more, the evaluation of the normalization constant in Equation (12) is dependent on the choice of minimum mass of 0.08 M_⊙; see the discussion in Section 3.1.4.)

In Figure 2, we plot the total mass of stars per Mpc³ that formed in a 50 Myr time-step at the time shown, M_formed: this is done by multiplying the SFR at a given time, $\dot{{\rho }_{* }}$, by our time-step of 50 × 10⁶ yr:

$$\begin{eqnarray}&&{M}_{\mathrm{formed}}=\dot{{\rho }_{* }}\times 50\times {10}^{6}.\end{eqnarray} \tag{ 13 }$$

The resulting values are shown as the top curve in Figure 2 (the solid red line). This curve shows that, as previously stated, the peak in SFR occurred at a time approximately 10.5 Gyr ago, when the universe was approximately 3.3 Gyr old.

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The second curve from the top in Figure 2 (red, dashed line) shows the total mass of stars per Mpc³ that still survive today from each time-step. This is formulated in the following way: first, the relationships in Equation (10) are used to express stellar mass as a function of main-sequence lifetime, $M\left({\tau }_{\mathrm{MS}}\right)$, then this is used in Equation (12) to arrive at an expression for the fraction of stellar mass formed during starbursts at some time in the past and that still survives today, f_survived(L), as a function of this lookback time. Multiplying the function shown by the top curve by this survival fraction, we obtain the total mass of stars that still survive today from the starburst at that time in the past,

$$\begin{eqnarray}&&{{M}_{\mathrm{survived}}=f}_{\mathrm{survived}}(L)\times {M}_{\mathrm{formed}}\end{eqnarray} \tag{ 14 }$$

which is shown by the second curve (red, dashed line). Hence, for instance, at a time 5 Gyr in the past, some 2.44 × 10⁶ M_⊙ Mpc⁻³ of stellar mass was formed during a 50 Myr time-step, of which some 1.51 × 10⁶ M_⊙ Mpc⁻³ survives today: a mass survival fraction of 62%.

If we again employ the Salpeter IMF for the total stellar mass, but use M_formed to serve as the M_total in Equation (11), we can evaluate a new value for the normalization constant, b. This will then be worked back into the Salpeter IMF expression (Equation (6)), allowing the number of stars formed per Mpc³ per 50 Myr time-step to be generated. The function expressing the fraction of stars that have survived since a starburst at a certain lookback time is then employed to create an accompanying plot of the number of these stars (which were formed during a particular 50 Myr time-step) that still survive today. The resulting plots are shown in Figure 2 as the two lower curves (blue): the blue solid line represents total number of stars formed per Mpc³ per 50 Myr time-step, while the blue dashed line represents the number of those stars that still survive today.

Note that, in order to create the blue plots in Figure 2, it was necessary to properly normalize the functions used in the preceding red plots, which were based on SFR data averaged over the universe, and we cannot be sure that this represents the SFR within the Milky Way fairly. Hence, we evaluated the area under the blue curves, and scaled the functions to properly represent the total number of stars in the Milky Way (over which there is a considerable range in estimated values, and we adopt an approximation of 250 billion stars¹ ) and the approximate volume of the Galaxy (which we model as a uniform cylindrical thin disk of volume 226 kpc³; see Rix & Bovy 2013).

For example, Figure 2 shows that, during a 50 Myr time-step, at a time 5 Gyr ago, a total of 1.26 × 10⁶ stars were formed throughout the the Milky Way as a whole, of which 97% survive today. In fact, we find that most stars formed still exist today, even if a larger fraction of stellar mass has been recycled. This is due to our assumption of the Salpeter IMF.

Note also that the red and blue curves in Figure 2 demonstrate an important point about the calculation of f_L: the total mass of stars formed in starbursts in the past has decreased today by a substantial fraction but, since the vast majority of stars are low-mass, and these have the greatest main-sequence lifetimes, the actual number of stars that survive today has decreased by a small amount. Therefore, the fraction of stars surviving today that are older than 5 Gyr, f_L, should be close to 100%. However, this calculation also allows us to determine the average age of the stars, which we use in Section 3.2.

As a final stage in this section, we create a plot showing the age distribution of all stars in the Galaxy today, (i.e., a plot of the cumulative number of stars surviving until today, versus the time since their formation) by performing a numerical integral of the bottom (blue dashed) plot in Figure 2, using the 50 Myr time-steps. The resulting cumulative plot is shown in Figure 3.

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Figure 4 shows the fraction of all stars surviving in the Milky Way today that are older than 5.0 Gyr: ${f}_{L}={0.963}_{-0.183}^{+0.034}.$

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This estimate is based on using the central values of the fitting constants, given in Table 2. This table also presents the corresponding values of f_L that come from the use of the upper and lower values of the fitting constants, allowing for an estimate of the uncertainty in f_L to be made.

In this section, we calculate the value of f_L by an independent method using the Chabrier IMF for comparison. We consider the Chabrier IMF for individual (nonmultiple) stars (Chabrier 2003), described in the following way:

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{dN}}{{dM}}=0.158\left(\displaystyle \frac{1}{\mathrm{ln}\left(10\,M\right)}\right)\\ \quad \times \ \exp \left(-\left(\displaystyle \frac{{\left(\mathrm{log}\left(M\right)-\mathrm{log}\left(0.08\right)\right)}^{2}}{2\times {0.69}^{2}}\right)\right)\ \mathrm{for}\ M\lt 1\,{M}_{\odot }\\ \displaystyle \frac{{dN}}{{dM}}={\mathrm{km}}^{-\gamma }\ \mathrm{for}\ M\gt 1\,{M}_{\odot }\end{array}\end{eqnarray} \tag{ 15 }$$

where γ = 2.3 ± 0.3, and k is chosen to provide a smooth transition between the two regions.

According to our calculation, the value of f_L, based on the Chabrier IMF, is approximately: f_L,Chabrier ≈ 97%. Hence, we conclude that the choice of the IMF has a low impact on our overall accuracy in f_L. This is due to the number of stars being dominated by the lowest-mass systems—the M dwarfs in particular.

It is worth discussing some of the implications concerning the result that 97% of the stars in the Milky Way are older than 5 Gyr. First, we find that the somewhat arbitrary assumption that CETI becomes established when the star is 5 Gyr old will not have a significant impact on our final result on the number of civilizations within the Galaxy, since there are relatively few stars younger than 5 Gyr: for instance, our calculation based on the critical time of 4.5 Gyr yields a value of f_L ≈ 97.4%, whereas for 5.0 Gyr we find of f_L ≈ 96.3%. Upon repeated calculation, as we vary this assumption of the time required for life to be established over the range 3.0–5.0 Gyr, we find a value of ${f}_{L}=97.5{ \% }_{-1.2 \% }^{+1.2 \% }$, showing only a small deviation from the estimate of 97%, which is based on the 5 Gyr assumption. This part of our calculation shows that the vast majority of stars in the Milky Way are in principle old enough to develop life as has occurred on Earth. The solar system formed late in the history of the Galaxy and most stars within the Milky Way are older than it.

The choice of minimum stellar mass at 0.08 M_⊙, taken as the minimum mass required for hydrogen fusion, may be expected to have a greater impact, given the large abundance of stars at the lower end of the mass range. However, by modifying the assumption of minimum stellar mass (over the range 0.06 M_⊙–0.10 M_⊙), we recalculate f_L, and our model appears to be quite insensitive to the choice of minimum stellar mass:

$$\begin{eqnarray*}&&{f}_{L}=96.27{ \% }_{-0.02 \% }^{+0.02 \% }.\end{eqnarray*}$$

Having developed an estimate of the fraction of all stars in the Milky Way that are older than 5 Gyr, it now remains to consider the issue of their metallicity, Z. Being older than 5 Gyr does not necessarily mean that a star is a likely target in which to search for life, as it may be a very old Population II star with low metallicity, which could presumably rule out the presence of rocky planets and life-forms. What is required is an investigation into the MDFs of stars throughout the Galaxy, so that we may evaluate an estimate of f_M: that is to say, the fraction of all Milky Way stars with a metallicity greater than some reference value. Using this as part of our criteria is an important aspect and again relates to the Copernican Principle: life on Earth has formed in a very metal-rich environment, thus it is seems likely that life would normally form in a metal-rich environment on other planets. One could argue a priori that having a stellar environment with metallicity that exceeds a certain reference value could be a prerequisite for the formation of habitable planets and even life itself. This is an assumption we make hereafter, but later discuss other possibilities.

According to Johnson & Li (2012), a suitable minimum stellar metallicity required for the formation of planets with Earth-like characteristics has been posited as 0.1 Z_⊙. However, for the present work, we employ three reference values in the investigation: 0.1 Z_⊙, 0.5 Z_⊙ and 1.0 Z_⊙ and explore how the results would vary within these assumptions.

Hayden et al. (2015) present details of their investigations into MDFs throughout different regions of the Milky Way disk, in the form of skewed Gaussian distributions, with particular values of mean, standard deviation, and skewness, as recalculated for our purposes in Tables 6–8 below. These are used to generate a distribution, and then the required percentages of stars above the three reference values of metallicity are determined.

Hence, for example, by using these measurements, it is possible to arrive at the following estimates for the Galactic region that contains the Sun (which we refer to as Region D; see Table 6):

Fraction of stars in Region D with metallicity greater than the reference value $1.0\ {Z}_{\odot }:{f}_{M,1.0}={0.5030}_{-0.0025}^{+0.0025}.$

Fraction of stars in Region D with metallicity greater than the reference value $0.5\ {Z}_{\odot }:{f}_{M,0.5}={0.9360}_{-0.0007}^{+0.0007}.$

Fraction of stars in Region D with metallicity greater than the reference value $0.1\ {Z}_{\odot }:{f}_{M,0.1}\gt 0.9999.$

This process has been carried out for each of the MDFs detailed in Hayden et al. (2015); in Table 4, and Tables A1 and A2 in the Appendix, we show the calculated percentages of stars within each region of the Galaxy with a metallicity lower than the reference values 1.0 Z_⊙, 0.5 Z_⊙, and 0.1 Z_⊙ respectively. The stated uncertainties in the skewness value in each table are used to generate minimum and maximum estimates of these calculated percentages.

Table 4.  Calculated Percentages of Stars with Metallicity Lower Than the Reference Value of 1.0 Z_⊙, within Each Region of the Galaxy

Region	Distance to Galactic Center, R/kpc	Mean Metallicity/dex	Standard Deviation/dex	Skewness (together with its Uncertainty)	Calculated Percentages of Stars in this Region with Z < Reference Value: 1.0 Z⊙
	Distance from midplane, ∣z∣/kpc: 0.00 < ∣z∣ < 0.50
B	3 < R < 5	+0.23	0.24	−1.68 ± 0.12	${28.97}_{-0.79}^{+0.80} \%$
C	5 < R < 7	+0.23	0.22	−1.26 ± 0.08	${22.52}_{-0.43}^{+0.44} \%$
D*	7 < R < 9	+0.02	0.20	−0.53 ± 0.04	${49.70}_{-0.25}^{+0.25} \%$
E	9 < R < 11	−0.12	0.19	−0.02 ± 0.03	${73.76}_{-0.15}^{+0.14} \%$
F	11 < R < 13	−0.23	0.19	+0.17 ± 0.06	${88.19}_{-0.18}^{+0.19} \%$
G	13 < R < 15	−0.43	0.18	+0.47 ± 0.13	${98.97}_{-0.06}^{+0.06} \%$
	Distance from midplane, ∣z∣/kpc: 0.50 < ∣z∣ < 1.00
I	3 < R < 5	−0.33	0.32	−0.50 ± 0.11	${87.77}_{-0.46}^{+0.48} \%$
J	5 < R < 7	−0.18	0.29	−0.50 ± 0.09	${77.31}_{-0.56}^{+0.54} \%$
K	7 < R < 9	−0.02	0.25	−0.49 ± 0.06	${57.86}_{-0.44}^{+0.43} \%$
L	9 < R < 11	−0.23	0.21	−0.22 ± 0.10	${87.12}_{-0.34}^{+0.33} \%$
M	11 < R < 13	−0.27	0.19	+0.28 ± 0.11	${91.58}_{-0.28}^{+0.27} \%$
N	13 < R < 15	−0.33	0.19	−0.60 ± 0.39	${96.57}_{-0.42}^{+0.34} \%$
	Distance from midplane, ∣z∣/kpc: 1.00 < ∣z∣ < 2.00
P	3 < R < 5	−0.27	0.29	−0.48 ± 0.14	${85.23}_{-0.67}^{+0.62} \%$
Q	5 < R < 7	−0.33	0.29	−0.32 ± 0.13	${88.81}_{-0.53}^{+0.49} \%$
R	7 < R < 9	−0.27	0.28	−0.53 ± 0.06	${86.07}_{-0.26}^{+0.25} \%$
S	9 < R < 11	−0.27	0.25	−0.37 ± 0.13	${87.58}_{-0.50}^{+0.47} \%$
T	11 < R < 13	−0.38	0.23	−0.40 ± 0.21	${95.74}_{-0.32}^{+0.29} \%$
U	13 < R < 15	−0.43	0.17	−0.60 ± 0.73	${99.54}_{-0.14}^{+0.08} \%$

Note. Region D, containing the Sun, is shown with the asterisk * in bold.

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In order to calculate an estimate of the overall percentage of stars in the Galaxy with metallicities exceeding the reference values, we require an estimate of the proportion of all stars that are located within the different regions analyzed in the last section. To express this, we create simple exponential models for the decrease in number density of stars throughout the Galaxy, with increasing Galactic radius and increasing vertical distance from the midplane: n(R) and n(z) respectively.

We model the variations in stellar number density with the following functions:

$$\begin{eqnarray}&&\begin{array}{l}n\left(R\right)\alpha \ {e}^{\left(-\tfrac{R}{{h}_{R}}\right)}\\ n\left(z\right)\alpha \ {e}^{\left(-\tfrac{z}{{h}_{z}}\right)}\end{array}\end{eqnarray} \tag{ 18 }$$

where we can take the scale length, h_R, and scale height, h_z, from Bland-Hawthorn & Gerhard (2016):

h_R: 2.5 ± 0.4 kpc

h_z: between 220 and 450 pc (with a mean estimate of 335 pc).

The number density modeling functions in Equation (18) are normalized and plotted, and the percentages of all stars within each of the regions mentioned in Table 4 are estimated from measurements on these plots.

The results for the fraction of the number density of stars within each region of the Galaxy (according to the exponential decay models of number density, Equation (18)) are shown in Tables 5 and 6. Error bounds are generated from further plots of the functions which acknowledge the uncertainty range in the exponential scale length and scale height. Note that, in order to create fairly weighted fractions of the absolute numbers of stars within each region (given in Table 11), we need to take into account the fact that these regions have unequal sizes.

Table 5.  Fraction of Number Density (n) within Each Radial Region of the the Milky Way (over All Vertical Positions)

Radial Location/kpc	Fraction of Number Density of Stars within This Region: fn(R)	Disk Area Fraction: $\tfrac{\mathrm{Area}\ \mathrm{of}\ \mathrm{annulus}}{\mathrm{Area}\ \mathrm{of}\ \mathrm{disk}}$
R < 3	${0.6991}_{-0.0538}^{+0.0612}$	0.0400
3 < R < 5	${0.1660}_{-0.0188}^{+0.0113}$	0.0711
5 < R < 7	${0.0746}_{-0.0178}^{+0.0144}$	0.1067
7 < R < 9	${0.0335}_{-0.0116}^{+0.0111}$	0.1422
9 < R < 11	${0.0150}_{-0.0065}^{+0.0074}$	0.1778
11 < R < 13	${0.0068}_{-0.0035}^{+0.0044}$	0.2133
13 < R < 15	${0.0030}_{-0.0018}^{+0.0027}$	0.2489
Totals	1.0000	1.0000

Note. In order to generate the fairly weighted fractions of the absolute number of stars in each region (in Table 11), the disk area fraction will also be required, which is calculated (without associated uncertainties) as follows:

$$\begin{eqnarray*}&&\mathrm{Disk}\ \mathrm{area}\ \mathrm{fraction}=\ \tfrac{\mathrm{Area}\ \mathrm{of}\ \mathrm{annulus}\ \mathrm{between}\ \mathrm{radii}\ \mathrm{identified}}{\mathrm{Area}\ \mathrm{of}\ \mathrm{MW}\ \mathrm{disk}\ \mathrm{of}\ \mathrm{radius}\ 15\,\mathrm{kpc}}.\end{eqnarray*}$$

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Table 6.  Fraction of Number Density (n) within Each Vertical Region of the the Milky Way (over All Radial Positions)

Vertical Distance from Midplane/kpc	Fraction of Number Density of Stars within This Region: fn(z)	Disk Thickness Fraction = $\tfrac{\mathrm{Thickness}\ \mathrm{of}\ \mathrm{region}}{4\,\mathrm{kpc}}$
0 < ∣z∣ < 0.5	${0.7752}_{-0.1044}^{+0.1218}$	0.25
0.5 < ∣z∣ < 1	${0.1533}_{-0.0609}^{+0.0676}$	0.25
1 < ∣z∣ < 2	${0.0343}_{-0.0238}^{+0.0622}$	0.50
∣z∣ > 2	${0.0372}_{-0.0372}^{+0.1891}$	0.00
Total	1.0000	1.00

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Table 5 covers the radial variations across the Galaxy, and contains the calculated disk area fraction, which represents the proportion of the surface area of each annular region out of the total disk area (taken out to a radius of 15 kpc, in accordance with the data used).

Table 6 covers the variation in number density for the vertical variations and takes into account the fraction of the disk thickness covered by each region. To calculate this disk thickness fraction, we take the full disk thickness as 4 kpc (2 kpc above and below the midplane) which (given the scale height we are using, h_z = 0.335 kpc), represents nearly six scalenheights, and hence will encompass approximately 99.8% of all of the stars, according to the exponential decay model.

As we have seen, the entire Galaxy is divided into 28 annular regions (labeled A to £, see Table 7), and the disk area fraction and disk thickness fraction each region occupies are used in conjunction with the number density proportions (n) to calculate the fraction of the absolute number of stars (N) which reside within each region, which is shown (together with error bounds) in Table 7.

The data from Table 4, those given in the Appendix and Table 7 can be combined to form weighted averages of the fraction of all stars within the Galaxy with metallicities which exceed the three reference values.

This process of creating a weighted average for the fraction of all stars in the Galaxy which exceed the given reference value yields the following results:

calculated result (reference value 1.0 Z_solar): f_M,1.0 = ${0.4982}_{-0.1617}^{+0.2522}$, therefore: 0.3365 < f_M,1.0 < 0.7504,

calculated result (reference value 0.5 Z_solar): f_M,0.5 = ${0.8172}_{-0.2875}^{+0.1828}$, therefore: 0.5297 < f_M,0.5 < 1.0000,

calculated result (reference value 0.1 Z_solar): f_M,0.1 = ${0.9738}_{-0.3723}^{+0.0262}$, therefore: 0.6015 < f_M,0.1 < 1.0000.

Again we find that even in the most pessimistic case in which we assume that the development of life requires a metallicity equal to that of the Sun (which we know has supported life on Earth) we still find that more than one third of stars have sufficient metallicity to potentially support life.