Age Spreads and the Temperature Dependence of Age Estimates in Upper Sco

The initial membership list of Upper Scorpius and spectral type measurements is compiled from Rizzuto et al. (2015), Pecaut & Mamajek (2016), and the compilation of Luhman & Mamajek (2012), which included previous surveys of Preibisch & Zinnecker (1999), Ardila et al. (2000), Preibisch et al. (2002), and Slesnick et al. (2006). Photometric data for low-mass members of Upper Sco are obtained from the 2MASS JHKs survey (Skrutskie et al. 2006), WISE W1, W2, W3, W4 survey (Cutri et al. 2013), and APASS (Henden et al. 2012). Proper motions in this work are compiled from GAIA DR1 (Gaia Collaboration et al. 2016), UCAC4 (Zacharias et al. 2013), PPMXL (Roeser et al. 2010), and SPM4 surveys (Girard et al. 2011). We focus on members with spectral types that range from F0–M5.

The Gaia Collaboration et al. (2016) DR1 TGAS astrometric catalog includes parallaxes of 126 F, G, and K-type objects that were previously identified as members of Upper Sco. They are located between 234° and 253° in R.A., and −34° to −14° in decl. Figure 1 shows that the distance distribution of these members has a mean of 144 pc, with a standard deviation of 17.6 pc, a mean error of 8.6 pc, and a systematic error of ∼6 pc (Arenou et al. 2017). A quadratic subtraction of these two errors indicates that the distance spread is ∼15 pc, consistent with previous estimates (e.g., Slesnick et al. 2008). Three stars with distances smaller than 100 pc (10 mas in parallax) or larger than 200 pc (5 mas in parallax) are rejected as members (Table 1). We adopt distances from the Gaia DR1 parallaxes when available, or the average distance of 145 ± 15 pc for stars without parallax measurements.

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Table 1.  Rejected Members Included in Pecaut et al. (2012)

2MASS	SpT	${\pi }_{{GAIA}}(\mathrm{mas})$
J15545986-2347181	G3V	13.08 ± 0.68
J15582930-2124039	F0IV	3.45 ± 0.26
J16123604-2723031	K4	1.03 ± 0.88

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Some members of the older Upper Centaurus-Lupus subgroup, a related association also within the Sco-Cen OB Association, likely contaminate this membership list. The related moving groups have similar space motions and overlap on the sky (de Zeeuw et al. 1999; Pecaut et al. 2012). Some contamination is also expected from the younger ρ Ophiucus star-forming region.

Figure 2 shows the proper motions of the candidates in both directions, where ${\mu }_{\alpha * }\equiv {\mu }_{\alpha }\cos \delta$. The average proper motion of all Upper Scorpius members considered in this paper is ${\mu }_{\alpha * }=-10.45$ mas yr⁻¹ and ${\mu }_{\delta }=-23.21$ mas yr⁻¹, with a total standard deviation of 6.41 mas yr⁻¹. To reject objects that are moving inconsistently with the common streaming motion, all stars with proper motions that deviate 3σ from the mean are rejected as members. The remaining 767 members are kinematic selected members and will be used in the following analysis.

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In this section, we describe how we combine spectral-type information from the literature with existing photometry to measure stellar properties and identify stars with disks, which are subsequently excluded from analysis.

2.2.1. Extinction

We adopt the conversion between spectral type, temperature, and color scales from Pecaut & Mamajek (2013). Uncertainties in temperatures are assigned to be 200 K at temperatures $\gt 4100$ K, 125 K at 3750–4100 K, and 50 K below 3750 K. Following Pecaut & Mamajek (2016), for stars with V-band photometry, extinction is estimated from the color excesses $E(V-J)$, $E(V-H)$, $E(V-K)$ relative to the expected photospheric color. For other sources, the extinction is estimated from the $E(J-K)$ color excess. The color excesses are then converted to extinction using A_J/A_V = 0.27, A_H/A_V = 0.17, and A_K/A_V = 0.11 (Fiorucci & Munari 2003). The adopted V-band extinction is the mean value of A_V estimated from these colors.

The top panel of Figure 3 shows the J − K color for low-mass members of Upper Sco. The black solid line is the relation between effective temperature and intrinsic color, which is derived from (Pecaut & Mamajek 2013), and the dashed black line is the color reddened by A_V = 2 mag. The median extinction is ${A}_{V}\sim 0.60$ mag. All objects reddened by ${A}_{V}\gt 2$ mag are excluded from further analysis. All extinctions ${A}_{V}\lt 0$ mag are set to A_V = 0 mag. Use of negative extinctions would have a negligible affect on the empirical isochrone of Upper Sco. The median A_V is constant versus spectral type to within 0.25 mag, which leads to systematic errors of $\lt 7$% in luminosity.

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2.2.2. Disk Identification

Disks produce excess emission in the infrared from dust, and in the optical from accretion onto the star. Because correcting photometry for these processes is challenging, we instead identify and exclude all disk sources in our sample. In this work, disks are identified based on a mid-IR color excess, with $E(K-W3)\gt 0.5$ (see also, e.g., Luhman & Mamajek 2012), as shown in Figure 4. These results are consistent with the disk identification in Upper Sco by Luhman & Mamajek (2012) and Rizzuto et al. (2015). Uncertainty in the $K-W3$ color leads to a higher rejection rate of diskless stars at $\lt 3000$ K.

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2.2.3. Luminosity

Luminosities are calculated from 2MASS J-band photometry and the bolometric correction of Pecaut & Mamajek (2013), obtained for the appropriate spectral type as follows:

$$\begin{eqnarray}&&\mathrm{log}\displaystyle \frac{L}{{L}_{\odot }}=-0.4\left({m}_{J}-5\mathrm{log}\displaystyle \frac{d}{10\,\mathrm{pc}}+{{BC}}_{J}-{M}_{\mathrm{bol},\odot }\right)\end{eqnarray} \tag{ 1 }$$

where the bolometric luminosity of the Sun, ${M}_{\mathrm{bol},\odot }$, is 4.755 mag (Mamajek 2012). The error of luminosity, $\sigma L$, is estimated as

$$\begin{eqnarray}&&\sigma \left(\mathrm{log}\displaystyle \frac{L}{{L}_{\odot }}\right)=0.4\sqrt{\sigma {({m}_{J})}^{2}+{\left[\displaystyle \frac{5\sigma (d)}{d\mathrm{ln}10}\right]}^{2}+\sigma {({{BC}}_{J})}^{2}}.\end{eqnarray} \tag{ 2 }$$

Estimating the first two terms is straightforward, as σ(m_J) is the observation error of the J band magnitude and σ(d) is distance spread (15 pc, see Section 2). To estimate the errors of bolometric corrections, we perform 1000 Monte Carlo simulations for each star. In each simulation, we assign the same error in temperature as discussed in previous sections, and derived a set of bolometric corrections for each star. The standard deviation is the adopted error in the third term in Equation (2). The error in bolometric magnitude then propagates to the error in luminosity.

Figure 5 shows the temperatures and luminosities of the diskless members of Upper Sco in an HR diagram. The empirical isochrone is calculated by dividing the sample into $\mathrm{log}{T}_{\mathrm{eff}}$ ranges from 3.5 (3000 K) to 3.9 (7000 K) into 20 bins of ∼0.02 dex. Bins with fewer than three stars are ignored. All stars that are located either above the 0.1 Myr isochrone or below the main sequence are rejected as non-members. Errors of temperatures and luminosities are assigned as described above. For each star, the adopted value and uncertainty for age is evaluated from ages obtained in 10⁴ Monte Carlo simulations in which these 1σ uncertainties are randomly applied to the effective temperature and luminosity. The empirical isochrone is then defined as the median of all luminosities within the bin. As has been established in other studies (e.g., Pecaut et al. 2012; Herczeg & Hillenbrand 2015; Rizzuto et al. 2015; Pecaut & Mamajek 2016), the average age of hotter stars is older than that of younger stars.

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Figure 6 compares the empirical isochrone with stellar evolution tracks⁵ from Siess et al. (2000, hereafter Siess), Dotter et al. (2008, hereafter Dotter), and the magnetic tracks of Feiden (2016, hereafter Feiden). The isochrone of Upper Scorpius is not parallel to any of the theoretical isochrones, as should be expected for co-eval members of a stellar cluster (if pre-main sequence evolutionary models are accurate). The age measurement strongly depends on temperature in all models for pre-main sequence evolution (see median age of each spectral bin in Table 2). The empirical isochrone calculated here for Upper Sco is very similar to that of Pecaut & Mamajek (2016) at temperatures hotter than 4000 K, and is several tenths of dex fainter at cooler temperatures.

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This sample suffers from several biases. First, the low-mass stellar population is incomplete, with significant spatial biases. The disk fraction versus luminosity within a small range of subclasses indicates that the sample may be preferentially missing faint targets at spectral types later than M3. Early-type members were identified by de Zeeuw et al. (1999), using Hipparcos over a large area of the sky, whereas many searches for low-mass stars focus on smaller regions. The sample also suffers from contamination of stars that belong to nearby, related regions. Finally, our exclusion of disk sources may preferentially exclude some of the youngest sources in the region. None of these biases affects our comparisons of inferred ages of low-mass and intermediate-mass stars.

Stellar populations are simulated using the pre-main sequence evolutionary models of Sotter et al. (2008). A sample of 10⁴ stars with masses 0.1–2.0 ${M}_{\odot }$ are generated. Their masses are randomly selected from the Chabrier initial mass function, which describes Galactic disk stars (Chabrier 2003). We then introduce to each star (or binary system) an age randomly generated from the star formation history.

The main sources of uncertainty are errors in spectral types, photometry, and distance, as well as in unresolved multiplicity. In the previous section, we assign to each member in Upper Sco errors in both effective temperature and luminosity. The mean errors in $\mathrm{log}T$/K and $\mathrm{log}L/{L}_{\odot }$ are assumed to be in a Gaussian distribution with a $1\sigma$ scatter of 0.01 and 0.05 dex, respectively (see also a similar implementation in Jose et al. 2017).

Unresolved binaries will lead to incorrect effective temperatures and luminosities, resulting in inferred ages that are too young (e.g., Preibisch & Zinnecker 1999; Slesnick et al. 2008; Reggiani et al. 2011). To estimate the effect of binarity (higher-order systems are ignored), the effective temperatures of multiple systems are assumed to be equal to that of the primary star, although the combined temperature will be lower than that of the primary alone (e.g., see discussion in Herczeg & Hillenbrand 2014). The luminosities of binary systems are estimated by summing the primary and companion luminosity, which will lead to small differences with our observed luminosity measurements from the J-band photometry.

We follow the quantification of the multiplicity fraction and mass ratio of Lu et al. (2013), which was estimated from the empirical results of Preibisch & Zinnecker (1999), Kouwenhoven et al. (2007), Kobulnicky & Fryer (2007), Lafrenière et al. (2008). The multiplicity fraction adopted here follows a power law with the form ${MF}(m)=0.44{m}^{0.51}$ given by Lu et al. (2013) for stars with masses less than 5 ${M}_{\odot }$, where m is the primary mass. All systems in our work have either one or two stars. The mass ratio $q\equiv \tfrac{{M}_{2}}{{M}_{1}}$ between the companion star and primary star also follows a power law,

$$\begin{eqnarray}&&p(q)=\left(\displaystyle \frac{1+\beta }{1-{q}_{{lo}}^{1+\beta }}\right){q}^{\beta }\end{eqnarray} \tag{ 3 }$$

where β ranges from −0.6 to 1.4. In this work, we adopt a flat companion mass ratio distribution (β = 0), following Reggiani & Meyer (2013).

The effect of pairing function is carefully discussed in previous work (see, e.g., Kouwenhoven et al. 2009). Here, we adopt the primary-constrained random pairing method, where the primary mass is derived from Chabrier IMF and the companion mass is determined by p(q). All companion stars with masses lower than $0.1\,{M}_{\odot }$ will not contribute to significant changes in the HR diagram, and are excluded from further analysis.

In this subsection, we describe how star formation history is implemented into our population synthesis. We first calculate stellar populations for the Smooth-Quench model, which has a constant star formation over some time period. We then calculate populations in the Smooth-Burst model, which adds a burst of star formation at the end of the Smooth-Quench model. The Smooth-Quench model is the simplest possible characterization of a star formation history. Although ad hoc additions to the Smooth-Quench model could become hopelessly complicated, adding a burst at the end of the Smooth-Quench model maximizes possible age differences between stars of different temperatures, thereby probing the limits of age differences versus spectral type. The Smooth-Burst model also may represent a reasonable scenario of a past supernova triggering a burst in star formation (see Section 4). The accelerating star formation model of Palla & Stahler (2000) is not investigated in this work, but would yield results that are intermediate between the Smooth-Quench and Smooth-Burst models.

These models are summarized in Figure 7. The Smooth-Quench model is described by two parameters: the time since the association first started forming stars, ${t}_{\mathrm{start}}$, and the time since star formation was quenched, ${t}_{\mathrm{quench}}$. The decrease in star formation after quenching is described by ${e}^{-{t}^{2}}$. The star formation rate is then analytically described by

$$\begin{eqnarray}\mathrm{SFR}(t)=\left\{\begin{array}{ll}1 & t\leqslant {t}_{\mathrm{start}}-{t}_{\mathrm{quench}}\\ {e}^{-{[t-({t}_{\mathrm{start}}-{t}_{\mathrm{quench}})]}^{2}} & {t}_{\mathrm{start}}-{t}_{\mathrm{quench}}\lt t.\end{array}\right.\end{eqnarray} \tag{ 4 }$$

In addition to ${t}_{\mathrm{start}}$ and ${t}_{\mathrm{quench}}$, the Smooth-Burst model adds a burst of strength A relative to the quiescent star formation rate at time t_quench. The star formation rate is then described as

$$\begin{eqnarray}&&\mathrm{SFR}(t)=\left\{\begin{array}{l}1\,\,(t\leqslant {t}_{\mathrm{start}}-{t}_{\mathrm{quench}})\\ A[t-({t}_{\mathrm{start}}-{t}_{\mathrm{quench}})]{e}^{-{[t-({t}_{\mathrm{start}}-{t}_{\mathrm{quench}})]}^{2}}\\ \,\,\,({t}_{\mathrm{start}}-{t}_{\mathrm{quench}}\lt t).\end{array}\right.\end{eqnarray} \tag{ 5 }$$

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For the Smooth-Quench model, by integrating over the star formation history, the ratio of the number of stars form at the quench/burst process N₁ and the number of stars form during the constant star formation rate N₀ is

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{1}}{{N}_{0}}=\displaystyle \frac{0.886}{{t}_{\mathrm{start}}-{t}_{\mathrm{quench}}},\end{eqnarray} \tag{ 6 }$$

and for the Smooth-Burst model it is

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{1}}{{N}_{0}}=\displaystyle \frac{0.5A}{{t}_{\mathrm{start}}-{t}_{\mathrm{quench}}},\end{eqnarray} \tag{ 7 }$$

where t_start and t_quench are both in the unit of Myr. Thus, if star formation occurs over 10 Myr (${t}_{\mathrm{start}}-{t}_{\mathrm{quench}}=10$), then an amplitude of the burst A = 20 would yield an equal number of stars formed during the burst and smooth periods of star formation.

In both of these constructions, the IMF, binary formation, and stellar evolution are assumed to be constant and independent of star formation rate. In our models, these parameters are calculated by generating 10⁴ stars over 10⁴ possible ages during the star formation history, then randomly assigning these stars a mass from the Chabrier IMF and binarity, as described above.

Our simulations are run with several sets of parameters, listed in Table 3. For each simulated population, the median ages of stars are estimated in different temperature ranges, with an empirical isochrone calculated following the same steps in Section 2. The empirical isochrones of the simulated HR diagrams from these models are then simplified into two measurable parameters: the age difference between F5 and M0 stars (∼6500 K and 3900 K, respectively) for the entire cluster and the luminosity spread of M0 stars. Binaries have only a minor effect in these simulations.

Table 3.  Parameters

	Smooth-Burst	Smooth-Quench
${t}_{\mathrm{start}}$	12, 14, 16, 18, 20	12, 14, 16, 18, 20
${t}_{\mathrm{quench}}$ (Myr)	2, 4, 6, 8, 10	2, 4, 6, 8, 10
A	20, 40, 60, 80	⋯

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3.3.1. Age Differences versus Photospheric Temperature

When star formation occurs over some duration, hotter stars will have older ages than cooler stars. Figure 8 shows the age versus temperature for a cluster that started forming stars 16 Myr ago and ceased forming stars 4 and 8 Myr ago. For this plot, the start time of star formation is fixed at ${{\rm{t}}}_{\mathrm{start}}=16\,\mathrm{Myr}$, with the quench time ${t}_{\mathrm{quench}}$ fixed at 4 Myr (top panel) and 8 Myr (bottom panel). Figure 9 shows these same differences, but parameterized by the age difference between F5 and M0 stars to show results from additional models. The time since the last star formation, ${t}_{\mathrm{quench}}$, is fixed in each of the panels.

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In the most basic Smooth-Quench model, the age differences between F5 and M0 stars are largest when star formation occurred recently and decreases with time. The recent star formation maximizes the number of intermediate-mass stars that are still fully convective. If no star formation has occurred in 8 Myr, then the differences between ages of F5 and M0 stars would be $\sim 2\mbox{--}3\,\mathrm{Myr}$. The duration of star formation (${t}_{\mathrm{start}}-{t}_{\mathrm{quench}}$) increases these differences. In the limit where ${t}_{\mathrm{start}}-{t}_{\mathrm{quench}}=0$, there would be no difference in age between F5 and M0 stars because all star formation would occur instantaneously.

Moving to the burst model, a modest burst (A = 20) increases the age difference by ∼4 Myr. Much larger bursts do not increase this effect significantly. Again, as ${t}_{\mathrm{quench}}$ increases, the intermediate-mass stars born during the burst will have enough time to evolve into radiative phases, thereby reducing any apparent age differences between F5 and M0 stars. Larger A-values will decrease the median ages of hot and cold stars. When ${t}_{\mathrm{quench}}$ is large (long duration since the burst), larger A values will result in smaller age differences.

The age differences become larger as the duration of star formation, ${t}_{\mathrm{start}}-{t}_{\mathrm{quench}}$, increases. For fixed ${t}_{\mathrm{start}}$ and A, larger ${t}_{\mathrm{quench}}$ will result in smaller age difference. For a single burst, we should not expect a large age difference between stars of different spectral type. A lengthy star formation process (${t}_{\mathrm{start}}-{t}_{\mathrm{quench}}$) is the most significant factor contributing to the spectral type-dependent age measurement.

3.3.2. Luminosity Spread

Any age spread during star formation will also lead to a spread in observed luminosities. In this subsection, we describe how different parameters in Smooth-Quench and Smooth-Burst models affect the luminosity spread of M0 type stars. Earlier spectral types are affected by stellar evolution and thus a poor probe of age spread with additional analysis, whereas M0 stars are fully convective and contract along the Hayashi track for the duration of the first 20 Myr of evolution simulated here.

Figure 10 show the effects of A and ${t}_{\mathrm{start}}$ on the luminosity spread. The luminosity spreads are always largest when star formation is recent, because the luminosity of an M0 star decreases with log age, whereas we are measuring age spread in linear space. In the Smooth-Quench model, for example, the age spread (standard deviation of all ages) will be $\sim 0.29\times ({t}_{\mathrm{start}}-{t}_{\mathrm{quench}})$. Although this age spread is constant, the luminosity spread decreases with age.

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All other factors are second-order effects in the luminosity spread. A modest burst (A = 20) will increase the luminosity spread. In a large burst (A = 80), the number of cool stars born during the burst overwhelm the number of stars formed during the smooth star formation, thereby reducing the age spread.

Our simulations of the star formation history of a cluster correspond to smooth star formation with no trigger (the Smooth-Quench model) and smooth star formation followed by a supernova trigger (Smooth-Burst model). Figure 11 compares the results of these simulations to the stellar population in Upper Sco for the Smooth-Burst model with parameters ${t}_{\mathrm{start}}=16\,\mathrm{Myr}$, ${t}_{\mathrm{quench}}=4\,\mathrm{Myr}$, and A = 60, and for the Smooth-Quench model with ${t}_{\mathrm{start}}=16\,\mathrm{Myr}$ and ${t}_{\mathrm{quench}}=4\,\mathrm{Myr}$.

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The Smooth-Quench model exhibits a temperature-dependent age measurement up to ∼6 Myr between KM stars and F stars, thereby accounting for half of the ∼12 Myr discrepancy (see Table 2). This simplest model demonstrates that an age spread of more than a few Myr in a young ($\lt 15$ Myr) population will lead to a significant temperature dependence in age estimates. This qualitative result is required by the evolution of intermediate-mass stars from the Hayashi track to the Henyey track. Cool stars are young and have a median age that changes slowly. For stars with $\mathrm{log}{T}_{\mathrm{eff}}/K=3.6\mbox{--}3.7$, the median age increases rapidly. At temperatures higher than $\mathrm{log}{T}_{\mathrm{eff}}/K\gt 3.7$, stars tend to be older and have median ages that change slowly. These simulation results are consistent with the age spreads of 4–5 Myr over a relatively large temperature range of cool stars (e.g., Herczeg & Hillenbrand 2015), and with the ∼3 Myr age spread for G type and F type stars (Pecaut et al. 2012).

Figure 11 shows that the Smooth-Burst model may explain entirely the empirical temperature-age relationship of Upper Sco. The burst increases the population of cool stars at the end of the star formation process, thereby maximizing the possible differences in ages between cool and hot stars.

Some of the age difference between K-M type and F type stars may be attributed to the incomplete physics in the calculations of pre-main sequence evolution of low-mass stars (e.g., Feiden 2016), although these explanations may introduce problems into other clusters. However, the age difference between G type stars (9 ± 3 Myr) and F type stars (13 ± 1 Myr), as calculated by Pecaut et al. (2012), are not easily explained by the same changes that are invoked to move the K-M type stars to older ages. The Smooth-Burst model recovers this age difference between G and F type stars without invoking any shortcomings in stellar evolution models or empirical temperature and luminosity measurements.

Figure 12 compares the simulated temperature-dependent luminosity spread to the measured luminosity spread within each temperature bin. The luminosity spread in the simulations and observations are consistent at the level of 1σ. The luminosity spread in the simulation peaks at slightly cooler temperatures than the observed luminosity spread, and also yields luminosity spreads that are systemically smaller than the measured spreads. Both of these uncertainties may be caused by observational errors, including sample incompleteness and mistaken membership, and also by errors in the stellar evolution models. A full evaluation of the star formation history will require a more complete membership list that is free of any biases with spectral type.

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Star formation in Upper Sco may have been triggered ∼5 Myr ago by the shock wave from a supernova explosion in Upper Centaurus Lupus (de Geus 1992; Preibisch & Zinnecker 1999). This star formation may have then been halted by a shock wave from another supernova that exploded in Upper Sco, destroying the molecular cloud. About 1 Myr ago, this shock wave would have arrived at the ρ Oph molecular cloud and triggered star formation. The scenario of the shock wave from Upper Centaurus Lupus triggering star formation in Upper Sco is challenged by the ∼11 Myr old age of Upper Sco, as measured from intermediate- and high-mass stars (Pecaut et al. 2012; Mamajek et al. 2013), but is consistent with the younger age estimated for low-mass stars. The conflict in the age measurements of different spectral types in the same cluster is an impediment to our understanding of star formation history.

In this work, we find that when a lengthy star formation is introduced before the trigger event, stars of different spectral types give different age estimations. In this context, we speculate about a plausible revision to the sequentially triggered star formation proposed by Preibisch & Zinnecker (1999), with a possible scenario outlined as follows (see also Figure 13): the star formation in Upper Sco started about 16 Myr ago, with a constant star formation rate over time. About 12 Myr ago, the most massive star in Upper Centaurus Lupus exploded, creating a shock wave that arrived at Upper Sco 5 Myr ago, triggering a burst of star formation. About 3 Myr later, the most massive star in Upper Sco may have destroyed the parent molecular cloud quenched star formation. This star would need to have formed early during the initial constant phase of star formation, because the formation of a 20 ${M}_{\odot }$ star, which explodes in ∼10 Myr, is far more likely than a 60 ${M}_{\odot }$ star, which explodes in ∼3 Myr (Ekström et al. 2012). The shock wave from this supernova explosion may have then arrived at ρ Oph, triggering star formation there. Both supernova and hot star winds are expected to have this type of positive and negative feedback, based on simulations (e.g., Dale et al. 2012) and also observations of clusters in massive star-forming complexes (e.g., Bik et al. 2012; Rivera-Ingraham et al. 2013; Jose et al. 2017).

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Compared with the scenario proposed by Preibisch & Zinnecker (1999), the similar history includes weak star formation with a long duration before the trigger event. This constant star formation may be related to that of Upper Centaurus Lupus and Lower Centaurus Crux, which have older average ages than Upper Sco (e.g., Pecaut & Mamajek 2016). The physics behind a lengthy star formation may be related to the cooling down of the molecular cloud. The depletion time, i.e., the lag time between the formation of molecular time and the start time of the burst of star formation, is an important timescale for the understanding of molecular cloud evolution. As a molecular cloud cools, star formation can occur in over-dense regions. This star formation is local, so its rate would be low. The over-dense regions may be distributed randomly over the molecular cloud, with small bursts that appear to constitute constant star formation over time. Although the estimation of depletion time requires larger samples and more detailed simulations, the temperature-dependent age estimate may provide a lower limit on past star formation.