# Stellar age determination - code

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I'm trying to determine the age of some stars. I have many parameters that characterize these stars: $$T_{eff}$$ ,log $${g}$$ ,$$[Fe/H]$$, $$V$$… I've tried to use the isochrones package but so far no success. I was wondering if any of you might have any recommendations!

I ended up using PARAM - http://stev.oapd.inaf.it/cgi-bin/param

Cheers,

THOT is an European Union Marie Skłodowska-Curie Action. Dr. Andrés Moya Bedón leads the project at the University of Birmingham under the supervision of Prof. William J. Chaplin. THOT is framed in the field of stellar dating and will develop a program to score the most precise stellar ages possible using the best knowing techniques, and considering the intro data uncertainties.

### Precise age determination of binary stars with a white dwarf component

by Dr. Andres Moya Bedon | March 13, 2018 | News | 0 Comments

### Empirical relations for the estimation of stellar masses and radii (accepted in ApJS)

by Dr. Andres Moya Bedon | March 13, 2018 | Results | 0 Comments

### The new Gaia Data Release 2 (DR2) is here

by Dr. Andres Moya Bedon | June 27, 2018 | News | 0 Comments

### Dr. A. Moya Bedón

He is a Marie Curie Fellow at the University of Birmingham from October-2017. He is the leader of the asteroseismic studies in the context of the first ESA S-Mission, CHEOPS, and he is a member of the scientific consortia of the space missions Plato (ESA) and Kepler (NASA). He is an active and passionate communicator, with a large record of outreach talks and papers.

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Research output : Contribution to journal › Article › peer-review

T1 - The age and stellar parameters of the Procyon binary system

N2 - The Procyon AB binary system (orbital period 40.838 yr, a newly refined determination) is near and bright enough that the component radii, effective temperatures, and luminosities are very well determined, although more than one possible solution to the masses has limited the claimed accuracy. Preliminary mass determinations for each component are available from Hubble Space Telescope imaging, supported by ground-based astrometry and an excellent Hipparcos parallax we use these for our preferred solution for the binary system. Other values for the masses are also considered. We have employed the TYCHO stellar evolution code to match the radius and luminosity of the F5 IV-V primary star to determine the system's most likely age as 1.87 ± 0.13 Gyr. Since prior studies of Procyon A found its abundance indistinguishable from solar, the solar composition of Asplund, Grevesse, and Sauval (Z = 0.014) is assumed for the Hertzsprung-Russell diagram fitting. An unsuccessful attempt to fit using the older solar abundance scale of Grevesse & Sauval (Z = 0.019) is also reported. For Procyon B, 11 new sequences for the cooling of non-DA white dwarfs have been calculated to investigate the dependences of the cooling age on (1) the mass, (2) core composition, (3) helium layer mass, and (4) heavy-element opacities in the helium envelope. Our calculations indicate a cooling age of 1.19 ± 0.11 Gyr, which implies that the progenitor mass of Procyon B was 2.59+0.44-0.26 M⊙. In a plot of initial versus final mass of white dwarfs in astrometric binaries or star clusters (all with age determinations), the Procyon B final mass lies several σ below a straight line fit.

AB - The Procyon AB binary system (orbital period 40.838 yr, a newly refined determination) is near and bright enough that the component radii, effective temperatures, and luminosities are very well determined, although more than one possible solution to the masses has limited the claimed accuracy. Preliminary mass determinations for each component are available from Hubble Space Telescope imaging, supported by ground-based astrometry and an excellent Hipparcos parallax we use these for our preferred solution for the binary system. Other values for the masses are also considered. We have employed the TYCHO stellar evolution code to match the radius and luminosity of the F5 IV-V primary star to determine the system's most likely age as 1.87 ± 0.13 Gyr. Since prior studies of Procyon A found its abundance indistinguishable from solar, the solar composition of Asplund, Grevesse, and Sauval (Z = 0.014) is assumed for the Hertzsprung-Russell diagram fitting. An unsuccessful attempt to fit using the older solar abundance scale of Grevesse & Sauval (Z = 0.019) is also reported. For Procyon B, 11 new sequences for the cooling of non-DA white dwarfs have been calculated to investigate the dependences of the cooling age on (1) the mass, (2) core composition, (3) helium layer mass, and (4) heavy-element opacities in the helium envelope. Our calculations indicate a cooling age of 1.19 ± 0.11 Gyr, which implies that the progenitor mass of Procyon B was 2.59+0.44-0.26 M⊙. In a plot of initial versus final mass of white dwarfs in astrometric binaries or star clusters (all with age determinations), the Procyon B final mass lies several σ below a straight line fit.

## Title: Estimation of distances to stars with stellar parameters from LAMOST

Here, we present a method to estimate distances to stars with spectroscopically derived stellar parameters. The technique is a Bayesian approach with likelihood estimated via comparison of measured parameters to a grid of stellar isochrones, and returns a posterior probability density function for each star's absolute magnitude. We tailor this technique specifically to data from the Large Sky Area Multi-object Fiber Spectroscopic Telescope (LAMOST) survey. Because LAMOST obtains roughly 3000 stellar spectra simultaneously within each

5-degree diameter "plate" that is observed, we can use the stellar parameters of the observed stars to account for the stellar luminosity function and target selection effects. This removes biasing assumptions about the underlying populations, both due to predictions of the luminosity function from stellar evolution modeling, and from Galactic models of stellar populations along each line of sight. Using calibration data of stars with known distances and stellar parameters, we show that our method recovers distances for most stars within

20%, but with some systematic overestimation of distances to halo giants. We apply our code to the LAMOST database, and show that the current precision of LAMOST stellar parameters permits measurements of distances with

40% error bars. This precision should improve as the LAMOSTmore » data pipelines continue to be refined. « less

1. Rensselaer Polytechnic Inst., Troy, NY (United States). Applied Physics and Astronomy Earlharm College, Richmond, IN (United States). Dept. of Physics and Astronomy
2. Chinese Academy of Sciences (CAS), Beijing (China). National Astronomical Observatories
3. Rensselaer Polytechnic Inst., Troy, NY (United States). Applied Physics and Astronomy
4. Univ. of Notre Dame, IN (United States). Joint Inst. for Nuclear Astrophysics (JINA)
5. Shanghai Astronomical Observatory (China)
6. Univ. of California, Santa Cruz, CA (United States). Dept. of Astronomy and Astrophysics, Lick Observatory
7. Chinese Academy of Sciences (CAS), Beijing (China). Nanjing Inst. of Astronomical Optics and Technology
8. Georgia State Univ., Atlanta, GA (United States). Dept. of Physics and Astronomy
9. Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)
10. Univ. of Utah, Salt Lake City, UT (United States). Dept. of Physics and Astronomy

### Citation Formats

5-degree diameter "plate" that is observed, we can use the stellar parameters of the observed stars to account for the stellar luminosity function and target selection effects. This removes biasing assumptions about the underlying populations, both due to predictions of the luminosity function from stellar evolution modeling, and from Galactic models of stellar populations along each line of sight. Using calibration data of stars with known distances and stellar parameters, we show that our method recovers distances for most stars within

20%, but with some systematic overestimation of distances to halo giants. We apply our code to the LAMOST database, and show that the current precision of LAMOST stellar parameters permits measurements of distances with

40% error bars. This precision should improve as the LAMOST data pipelines continue to be refined.>,
doi = <10.1088/0004-6256/150/1/4>,
journal = ,
number = 1,
volume = 150,
place = ,
year = <2015>,
month = <6>
>

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Confused of York writes:
I want to know the age of a particular star, which seems to be given in a reference I've found but I don't understand the units.
The star is HD 155902, the reference is

and the figure for the system's age is 9.27 +/- 0.30 (log Myr^-1). What is that in good old vanilla years?

Confused of York writes:
I want to know the age of a particular star, which seems to be given in a reference I've found but I don't understand the units.
The star is HD 155902, the reference is

and the figure for the system's age is 9.27 +/- 0.30 (log Myr^-1). What is that in good old vanilla years?

Concerning the actual question, it would help to know the method of age determination being referenced by the book/paper in question.
See for example: (2013) Chromospheric Activity as an Age Indicator

## 5. DISCUSSION

The stellar stream in NGC 5907 has been previously modeled with a satellite galaxy accretion event (Martínez-Delgado et al. 2008, total mass ratio 1:4000) and with a gas-rich major galaxy merger (Wang et al. 2012, total mass ratio 1:3–1:5). Neither N-body model achieved a quantitative fit to the stream, but both were able to qualitatively create features similar to those seen in the observations.

The main objection to the satellite galaxy accretion model is that there is no obvious remnant of the nuclear region of the disrupted satellite (Wang et al. 2012). In NGC 5387 Beaton et al. (2014) found signs of a very blue star-forming cluster right at the intersection of a stellar stream and the host galaxy disk. They speculated that this could be the starbursting nuclear remnant of the mostly disrupted satellite galaxy. However, no such star-forming region has been found near the intersections of the disk of NGC 5907 and the stellar stream, and we do not see any such blue star-forming region along the brightest section of the stellar stream that we have imaged. Nor have we detected any signs of active star formation in the stellar stream.

The main objection to the major galaxy merger model is that NGC 5907 appears to have a thin disk with no obvious bulge (see Athanassoula et al. 2016) and does not look like a severely perturbed major merger remnant (see major merger simulations by, e.g., Barnes 1992). In the gas-rich major merger scenario, the thin disk of NGC 5907 is rebuilt around a small bulge. A similar claim for the creation of the faint stellar stream features around NGC 4013 by a major merger has recently been made by Wang et al. (2015). Another objection to the major merger model could be the lack of any detected neutral atomic hydrogen (H i ) gas in the stellar stream (Shang et al. 1998) if it is interpreted as being a long remnant tidal tail.

Our results suggest that the stream in NGC 5907 is relatively metal-rich: the best fitting FSPS model has a metallicity log [Z/Z] (≈[Fe/H], assuming that the fractional percentage of hydrogen is the same as in the Sun and that the iron abundance and the total metal abundances are roughly the same) = −0.3. This should be compared to the reported metallicity in the Sagittarius stream around the Milky Way, from a peak [Fe/H] of −0.3 in its core to a median [Fe/H] from −0.7 to −1.1 in its leading arm (Chou et al. 2007). The Sagittarius stream is known to be an interacting satellite galaxy, and its core has been identified. While the bound mass of the Sagittarius dwarf (the progenitor of the Sagittarius stream) is currently about 2.5 × 10 8 M, the N-body modeling of Law & Majewski (2010) suggests an original mass of 6.4 × 10 8 M. Meanwhile, in the giant southern stellar stream of M31, the [Fe/H] has a strong peak at −0.3, with a mean of −0.55 and a median of −0.45 (Guhathakurta et al. 2006 Kalirai et al. 2006 Tanaka et al. 2010). The disrupted precursor galaxy for the giant southern stellar stream in M31 could have been a satellite of stellar mass 4 × 10 9 M (Dekel & Woo 2003 Tanaka et al. 2010). A comparison to the gi color of about 0.6–0.8 for the umbrella in NGC 4651 (Foster et al. 2014) that is supposedly a result from a merger of a metal-poor satellite suggests that the merging galaxy in the NGC 5907 system was more metal-rich.

Using the metallicity of the NGC 5907 stellar stream and the metallicity versus stellar mass relationship in Dekel & Woo (2003) and Kirby et al. (2013, reproduced in Figure 11 of the current paper), the disrupted companion of NGC 5907 would have had a stellar mass of about 1 × 10 10 M, with a large uncertainty that extends down to

1 × 10 9 M. Note that the mass–metallicity relationship evolves when going to a higher redshift sample. Similarly, Equation (9) in Kirby et al. (2011) predicts Ltot = 4.1 × 10 10 L for the disrupted companion of NGC 5907, using our derived [Fe/H] value of −0.3. This would make this system qualify as a major merger, because the dynamical disk mass in NGC 5907 has been estimated to be about 1.4 × 10 11 M (Casertano 1983, adjusted for the adopted distance of 17 Mpc in our study), of which gas may constitute about 10% (Dumke et al. 1997). Just et al. (2006) modeled the disk of NGC 5907 with a stellar mass of 2 × 10 10 M, making the merger an almost equal disk mass merger. However, using the conversion from IRAC 3.6 and 4.5μm flux densities by Eskew et al. (2012), the stellar mass of NGC 5907 is about 7.8 × 10 10 M. We adopt this as the best estimate to compare with our (partly) IRAC-based metallicity measurement.

Figure 11. Stellar mass–stellar metallicity relation, reproduced from Figure 9 of "The Universal Stellar Mass–Stellar Metallicity Relation for Dwarf Galaxies" by Kirby et al. (2013). The lower left shows the relation for local group dwarf galaxies and the upper right for more massive galaxies in SDSS from Gallazzi et al. (2005). The dotted line in the diagram for the more massive galaxies is the median of the stellar metallicity distribution as a function of stellar mass. The black arrow was added to point to the metallicity value measured in this work for NGC 5907. See Kirby et al. (2013) for more information.

We have estimated the mass in the IRAC-imaged part of the stellar stream (Table 1). The value we obtain, 2.1 × 10 8 M, is consistent with the value estimated by Martínez-Delgado et al. (2008), 3.5 × 10 8 M, because Martínez-Delgado et al. (2008) used the luminosity of the whole stream that is seen in Figure 1 to estimate the mass. This mass would imply a satellite accretion event that may also be consistent with our color findings if one allows for systematic issues in the stellar population modeling (see Section 4). On the other hand, the estimated stream mass is only a lower limit to the progenitor mass and thus it cannot accurately constrain the merger scenario, because in a minor merger most of the merged companion mass has presumably become part of the bulge and disk of the primary galaxy, NGC 5907.

## 5. Results

### 5.1. Best-Fit Model Without Frequency Shifts

We computed the best-fit model as described above for the two real stars and for the proxy stars using as input the spectroscopic parameters of Table 1 and, for the real stars, the observed frequencies and, for the proxies, the theoretical frequencies modified according to the description of section 2, respectively. Table 2 summarizes the results for the two stars and their proxies for the case where no frequency changes due to magnetic activity are added. The first row corresponds to KIC8006161 and the second one to its proxy. As it can be seen, the results have slightly lower errors for the proxy than for the actual star. The third row corresponds also to the proxy but in this case the frequency and spectroscopic errors were halved. As it can be seen in Table 2 the resulting uncertainties in the stellar parameters are reduced by the same order of magnitude. Finally, for KIC8006161 we have also carried out a similar fit but considering only a subset of modes with ℓ < 3 and frequencies in the central range, with the lowest frequency errors, namely, ℓ = 0 modes with n = 17�, ℓ = 1 modes with n = 15� and ℓ = 2 modes with n = 15� (the full frequency set was given in the first paragraph of section 2). The resulting model parameters are shown in the last row of Table 2, and we can conclude that the estimated uncertainties are similar to those in the first row where the full set of frequencies were considered.

Rows 5, 6, and 7 in Table 2 correspond to KIC9139163, its proxy and its proxy with halved frequency errors. Here the results are similar in all the cases. The most relevant differences between the star and its proxy are the χ 2 values, which are higher for the observations. This could be the result of the approximations and uncertainties of the physics considered in the evolution codes. Some of these discrepancies between models and observations can come from the magnetic activity and in principle one could wonder if frequency shifts induced by the magnetic activity could be detectable through higher χ 2 values compared to other stars of the same type.

### 5.2. Adding the Frequency Shifts

We then added the frequency shifts with the ℓ-dependent frequency shift and with the frequency dependent shift. The merit functions, χ 2 , are shown in Figure 5 where panel A (resp. C) corresponds to KIC8006161 (resp. KIC913916) with the shift that depends on ℓ and panel B (resp. D) corresponds to KIC8006161 (resp. KIC913916) where the shift varies with frequency. On the one hand, for the ℓ-dependent cases the X-axis corresponds to the difference in μHz between the radial and non-radial modes. For instance a value of 0.3μHz corresponds to a shift of δν = 0.2μHz for the radial oscillations and δν = 𢄠.1μHz for non-radial oscillations. In addition a constant shift was also included as indicated in section 3.2. The circles with error-bars at x = 0 correspond to shifts that only include the constant term whereas the crosses at x = 0 correspond to the results without frequency shifts.

Figure 5. Merit function χ 2 as a function of the frequency shift introduced for KIC8006161 (A,B) and for KIC9139163 (C,D). Here ℓ or ν indicates if the shift considered was ℓ or ν dependent. Red points correspond to adding the frequency shifts to the actual stellar data, green points correspond to adding the frequency shifts to the proxy with the same frequency errors and blue points correspond to the proxy with half of the observed frequency errors. The magnitudes shown in the X-axes are explained in section 3.1 and 3.2 (see also section 5.2).

On the other hand for the ν-dependent cases the X-axis is the amplitude A1 of the oscillatory function introduced in Equation (1), as factors of the actual values found for each star (see Figure 4 for a graphic representation of the frequency shifts corresponding to x-values of 1 in panels B and D of Figure 5). At x = 0 there are again two values per star: the circle which include the A0 term and the cross corresponding to the result without frequency shifts.

Let us consider the simulations in panel A. The red points correspond to the χ 2 values obtained by adding the frequency shifts to the actual stellar data and errors. In this case χ 2 increases with the absolute value of the frequency shift introduced but the changes are within errors and could hardly indicate an incorrect fit even for the highest values (frequency shifts of 0.45μHz). For the proxy star with the same frequency errors (green points) we obtain a similar behavior. However, for the proxy star with half the frequency errors (blue points) the χ 2 values clearly increase even when only the constant shift is considered for frequency shifts higher than 0.3μHz they are above the 1-σ level. These kind of frequency shifts is within the range of observed values as shown in section 3, but presumably they will be only present at a given time of the magnetic activity.

The results for the frequency-dependent shifts (panel B of Figure 5) show that for the extreme changes (that is an oscillatory frequency shift with an amplitude 5 times those observed in KIC8006161) even when considering the actual stellar parameters and errors (red points) the χ 2 increases significantly. For the proxy with half the errors (blue points) even amplitudes of 2.5A1 could lead to an incorrect model fit.

For KIC9139163 the results on the oscillatory dependent term are similar (panel D) to the first star, however for the ℓ-dependent case (panel C) it seems that an acceptable fit is always obtained.

As a conclusion, an analysis based on the merit function χ 2 cannot identify a bad model fit if we introduce a ν-dependent shift with an amplitude A1 for the oscillatory component smaller that 2.5 times the observed one. On the other hand, any value considered in this work for the ℓ-dependent shift can be identified as a bad fit, except for the ~ 1M case (KIC8006161) and assuming frequencies errors half the observed ones.

### 5.3. ℓ-Dependent Frequency Shifts

Despite the different χ 2 values found above, the stellar parameters derived for the proxy with the full errors and half errors gives very similar results, so from now on we will only show results for the proxy with half the actual errors.

We first derived the stellar parameters from the minimization procedure after introducing the ℓ-dependent frequency shift (see Figure 6 for KIC8006161). For some of the parameters the changes induced by the frequency shifts are higher than the formal uncertainties. In particular for the age we find a clear trend with a decrease of ~ 3.5% every 0.1μHz of increase in the difference between the frequency shifts of radial and non-radial modes. Thus, for such a star where different ℓ are experiencing different frequency shifts due to magnetic activity, the age estimate can be more than 10% away from the real age of the star. This can be compared with the uncertainties of 3% when using the actual frequencies and errors as well as the 1% uncertainty for the proxy with half frequency errors. Qualitatively, these results can be understood in terms of the associated change in the small separation that when using spherically symmetric models can only be interpreted in terms of changes in the stellar core. For the mass and radius the changes are smaller, of the same size than typical (formal) uncertainties. Specifically there is an increase of ~ 1% per 0.1μHz in the mass and of ~ 0.3% per 0.1μHz in the radius that are of the same order as the formal uncertainties given in Table 2.

Figure 6. Stellar parameters derived from the minimization procedure as a function of the ℓ-dependent frequency shift introduced in the frequencies. Red points are for KIC8006161 whereas blue points corresponds to the proxy.

Figure 7 shows the results when an ℓ-dependent frequency shift is introduced in the eigenfrequencies of KIC9139163. In this case, the frequency shifts do not change the results by more than the formal uncertainties, except perhaps for the overshooting parameter fov for which we find changes of 0.02, above the 1-σ level.

Figure 7. Stellar parameters derived from the minimization procedure as a function of the ℓ-dependent frequency shift introduced in the frequencies. Red points are for KIC9139163 whereas blue points corresponds to the proxy.

### 5.4. Frequency Dependent Shifts

Let us now consider the frequency dependent changes discussed in section 3. In Figure 8 we show the results for KIC8006161. Remember that in this case X-axis correspond to the amplitude A1 in Equation (1) in times the A1 value derived from the observations. There are two points with A1 = 0, the cross corresponding to the modes without any frequency shift (A0 = 0 in Equation 1) and the dot corresponding to a frequency shift with the constant coefficient A0 ≠ 0 derived by Salabert et al. (2018) but with the oscillatory term removed.

Figure 8. Stellar parameters derived from the minimization procedure as a function of the ν-dependent frequency shift introduced in the frequencies. Red points are for KIC8006161 whereas blue points corresponds to the proxy.

In this case, the changes in the age estimate are within the error bars. However, there is a clear increasing trend in the mass, radius, and initial helium abundance. Specifically by taking A1 equals to the observed value, the age decreases by 0.4%, the mass increases by ~ 0.6%, the radius by ~ 0.15%, and the helium abundance by ΔYini ~ 0.002 (1%). These numbers are in practice below typical errors and could hardly be discriminated from other source of errors. In principle for large oscillatory amplitudes (higher values of A1) the changes could be more relevant, ΔYini ~ 0.01 (5%). However, as discussed early, these larger changes would result in high χ 2 values, which would allow us to identify an incorrect model fit.

Figure 9 shows the results for KIC9139163. In this case the frequency shifts introduced do not change the results by more than the formal uncertainties though like KIC 8006161, similar trends in the age, mass, and radius can be seen.

Figure 9. Stellar parameter derived from the minimization procedure as a function of the ν-dependent frequency shift introduced in the frequencies. Red points are for KIC9139163 whereas blue points corresponds to the proxy.

## Oscillation mode variability in pulsating hot B subdwarfs and white dwarfs from Kepler photometry

The Kepler satellite provides unprecedented and uninterrupted high-quality photometric data with a time baseline of about 4 yr collected on pulsating stars, which is the unique opportunity to characterize the long-term behaviours of oscillation modes. A mode modulating in amplitude and frequency can be independently inferred by its fine structure in the Fourier space, detected by the sliding Lomb-Scargle periodogram, and measured by prewhitening the entire light curves parts by parts. We apply these techniques to the evolved compact stars KIC 3527751, KIC 08626021 and KIC 10139564 and find that many rotational multiplets whose components show clear amplitude and/or frequency variations. These modulations can be periodic, irregular and stable over the Kepler observations, which are the first signatures of nonlinear interactions due to the resonant mode coupling theory. Our results suggest that oscillation modes with diverse variations should be a common phenomenon in pulsating sdB and white dwarf stars. This resonate an idea that the closed peaks need to be seriously examined for mode identification, which is a key input parameter of seismic modelling. These various modulation patterns motivate more precise stellar oscillation theory to be developed. It also raise a warning to any long-term project aiming at measuring the rate of period change of pulsations due to stellar evolution or discovering stellar (planetary) companions around pulsating targets. These phenomena can also be thoroughly examined in many types of pulsating stars over the entire HR diagram from the photometry of the upcoming TESS mission.

The meaning of an isochrone is in the name. It's a set of stellar models compute at the same (iso) age (chrone). So, in practice, what this means is that some theorists use a stellar evolution code to compute evolutionary tracks (i.e. a star's properties over time, or age) for a number of different masses, and then they connect all the models of different masses at the same age.

Here's a figure from some of my teaching material that might help. (One could annotate this much better but this is from a lecture, so I'm standing there explaining. )

The grey lines show evolutionary tracks for stars with masses from 1 to 10 solar masses, in steps of 1 solar mass, up to core helium ignition (I think). But they all evolve at different rates, which isn't shown by the grey curves. (Roughly speaking, the main-sequence lifetime of a star goes like $M^<-5/2>$.)

The black lines are where the tracks have been interpolated at fixed ages, as indicated (roughly uniformly spaced in $log(mathrm)$). The first, left-most black line is the "zero-age main-sequence" (ZAMS), which is roughly where a star starts burning hydrogen in the core. The next isochrone (30 Myr) is the one closest to the ZAMS, where the most massive stars have already disappeared completely because they live for less than 30 Myr. Stars around 6-7 solar masses have started finishing off the hydrogen in their cores, but lower mass stars have barely budged.

As the isochrones get older, more and more stars "peel away" from the ZAMS. Note, however, that the very low-mass stars (less than about 1 solar mass) basically don't budge at all. These stars evolve very slowly.

In my experience, isochrones are these theoretical curves. However, they're closely related to cluster colour-magnitude diagrams (CMDs) because we expect the stars in a cluster to have been born at roughly the same time. Thus, when we look at them in a CMD (or Herzsprung–Russell diagram), we should be seeing stars with the same age, so they should look like an isochrone. By comparing the data to isochrones (with a suitable choice of the composition, which I haven't discussed), astronomers can estimate the ages of the clusters.