# Why is the track of the subgiant stage almost horizontal on the HR diagram?

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It is stated that:

After the main sequence, as fusion weakens or stops in the core, outward radiation weakens. The helium core contracts and heats up. Gravitational energy is converted to thermal energy again!

The star will appear to cool slowly and will undergo a modest increase in luminosity. During this phase, the path the star will follow in the HR diagram is almost horizontal to the right of its position on the Main Sequence. Stars in this phase are usually referred to as subgiants.

But I still can't understand.

I will replace the previous answer to focus on the "subgiant" branch prior to red giant, rather than the pre-main-sequence or the "horizontal branch" of core helium fusion. Those are other times that the luminosity is constant, but this question is about the subgiant branch, which I missed before.

The reason the luminosity is nearly constant on the subgiant branch is related to the "mass-luminosity relation" of pre- and main-sequence stars. It is due to radiative diffusion and how it leads to a luminosity that depends only on mass, for a given composition. If you compare to pre-main-sequence tracks, you should find that the subgiants more or less retrace that prior evolution, just with a somewhat higher luminosity because many of the electrons have been swallowed into neutrons, reducing the opacity and increasing the all-important rate of radiative diffusion. It's essentially just a helium-dominated mass-luminosity relation, instead of hydrogen-dominated, as the radius rises due to the details of how the interior is evolving.

The reason the luminosity eventually rises steeply on the red giant branch is that as the degenerate core starts to build in mass, it starts to control the temperature of the fusing region, and this significantly changes the internal structure in ways that get into explaining red giants.

From blackbody radiation ($$L = c r^2 T^4$$) the horizontal evolution implies increasing radius and decreasing temperature.

This happens because of the depletion of H at the core:

Depletion of H at core -> core contraction -> increasing core temperature + starting H fusion in the envelope (in HR diagram, the star goes up) -> to maintain the equilibrium from the injection of energy of H fusion in the envelope, envelope expands and cools (moving horizontally to the right in HR diagram) ->…

## The Transition to the Red Giant Phase for Sun-like stars

Whenever you are considering the physical state of a star, you should separately consider its core (where the temperature and pressure are very high) and its envelope (where the temperature and pressure are substantially lower). The core is where fusion occurs, but the envelope is what we observe, so we have to infer what is going on in the core by observations of the envelope.

The most important concept to recall when studying stars is the concept of hydrostatic equilibrium. When nuclear fusion is going on in a star's core, the pressure created by this process pushes outward and balances exactly the inward pull of gravity. The first stage of the evolution of a star is the Main Sequence stage, and this accounts for approximately 80% of the star's total lifetime. During this time, the star is fusing hydrogen in its core. The star's color (a measurement of its surface temperature) and luminosity only change slightly over the course of its Main Sequence lifetime as the rate of nuclear fusion changes as the star slowly converts hydrogen to helium. When the star initially begins fusing hydrogen it is said to be on the Zero Age Main Sequence (ZAMS). Over a star's Main Sequence lifetime, as it fuses hydrogen into helium, its outer envelope will respond to slow internal changes, so its position in the HR diagram is not completely fixed. For example, we expect our Sun to brighten and its color to vary slowly over its roughly 10 billion year lifetime on the Main Sequence. By the end of its Main Sequence lifetime, it will be approximately twice as luminous as it is now!

When any star has used up the majority of the hydrogen in its core, it is ready to leave the Main Sequence and begin its subsequent evolution. From here on out, we will be considering the post-Main Sequence evolution for different types of stars. During the Main Sequence phase, core hydrogen fusion creates the pressure (in the form of radiation pressure and thermal pressure) that maintains hydrostatic equilibrium in a star, so you should expect that when a star's core has become filled with helium and inert, the star will fall out of equilibrium. As the total pressure decreases, gravity will once again dominate, causing the star to begin to contract again. You should be able to predict that when a stellar core contracts, its temperature will increase. So the star will continue to generate energy in its core, even when core hydrogen fusion ends, through the gravitational contraction of the core. Although fusion has turned the hydrogen in the core into helium, most of the outer layers of the star are made of hydrogen, including the layer immediately surrounding the core. Thus, when the core reaches a critical density and temperature during its contraction, it can ignite hydrogen fusion in a thin shell outside of the helium core. The helium core will also continue to generate energy by gravitational contraction, too. If you think of the Main Sequence as the “hydrogen core fusion” stage of a star's life, the first stage after the Main Sequence is the hydrogen shell fusion stage. During this stage, the rate of nuclear fusion is much higher than during the Main Sequence stage, so clearly the star cannot stay in this stage as long. For a star like the Sun, it will only remain in this stage for a few hundred million or a billion years, less than 10% of the Sun's Main Sequence lifetime.

While these internal changes are occurring in the star, its outer layers are also undergoing changes. The energy being generated in the core will be more intense than during the core hydrogen fusion (Main Sequence) phase, so the outer layers of the star will experience a larger pressure. The increased pressure will cause the outer layers of the star to expand significantly. As a side effect of this expansion, the outer layers of the star will cool down because they are now farther away from the energy source (the hydrogen shell around the core). The observable changes in the outer layers of the star will occur in two phases. First, the star will appear to cool slowly and will undergo a modest increase in luminosity. During this phase, the path the star will follow in the HR diagram is almost horizontal to the right of its position on the Main Sequence. Stars in this phase are usually referred to as subgiants. Next, the star will grow to as much as, or even more than, 100 times its original size, which will cause a significant increase in luminosity with only a small decrease in temperature, so the star will move almost vertically in the HR diagram. Stars in this area of the HR diagram are usually referred to as red giants. The evolutionary track for the star as it undergoes the transition to a red giant is shown below:

If you look at the dashed lines in this HR diagram, they represent lines of constant radius. When a star has reached the tip of the red giant branch (the highest point in luminosity on the track above), it has a radius of approximately 100 solar radii. There are several well known red giant stars even larger than this, which have radii of several hundred solar radii. The immense growth expected in the Sun when it becomes a red giant will cause its radius to swell from roughly 1 AU to perhaps 2 AU or so. This means that Mercury and Venus will definitely be engulfed by the Sun, and the Earth and Mars are likely to be engulfed as well. The core of the Sun when its envelope is 1 AU will only be of order 10 Earth radii, or a factor of more than 2,000 times smaller than the radius of the envelope.

Compare the illustrations below: the first shows the Sun as a red giant and compares it to the Sun at its current size (Fig. 6.2), and the second shows the measured size of Betelgeuse (Fig. 6.3).

As I understand the process, it starts when the contraction of the star (after it has run out of hydrogen in the core) reaches a point where a large shell of hydrogen around the core ignites. The amount of fusion taking place in this shell is greater than what took place in the core, a result of the larger volume of the shell compared to the core. This greater energy output puffs up the outer layers of the star, turning it into a red giant.

Actually when I first learned stellar evolution two years ago, I directly skipped from main sequence to red giant, omitting subgiant, as most people regard it 'minor.

But I read some articles that says the ignition point of hydrogen shell is the turnoff point from the main sequence (i. e. Start of subgiant). Then my question is the difference between subgiant and red giants, as both of them fuse hydrogen shell.

For the second question, yes, it does decrease. But if you consider that it similarly burns hydrogen like subgiants, why does it have a near 90° turn off? If it can increase in radius so much without decreasing the temperature a lot (unlike subgiants), there must be vigorous heating event to do the effect?

I read wikipedia about 'dredging-up', where the fusion reach the very surface, and brings some metals up. But I don't know if it relates to my question.

Not sure. Wiki's article on subgiants says the following:

Yes indeed subgiants are going to be red giants eventually. The graph I showed is a model for a star slightly larger than sun. What I'm curious about is the quick turn off from subgiant branch to red giant branch.

For every other turn offs in the graph it has a trigger event: e. g.
Main sequnce-->subgiant: H shell ignition
Red giant-->horizontal branch: helium flash

I'm wondering what is the powerful event turning subgiant branch horizontal suddenly to a vertical red giant branch.

I've been trying to get a sensible answer out of Prialnik's book on stellar evolution, but I'm not sure I grok it.
What I think is happening during the subgiant phase, is the hydrogen depleted core contracts and heats up, which then expands the envelope without significant change in total energy production, or even with a decrease as fusion processes switch from core to shell burning, hence little to no luminosity change.
As the core (and by extension - shell) heats up, fusion in the shell becomes dominated by the CNO cycle, whose high temperature sensitivity means that any further temperature increase due to helium ash being deposited onto the core (i.e further collapse) leads to rapid increase in luminosity. The star expands to find new equilibrium. This would be the near vertical ascension on the H-R diagram.
That's the best I can make out of it. Furthermore, I'm not sure if this reasoning works for intermediate-mass stars, whose cores don't reach the Shonberg-Chandrasekhar instability. But then again, for those stars the subgiant phase is not such a horizontal line.

Maybe we can get @Ken G or @e.bar.goum to comment?

The shell burning would be CNO cycle even in the subgiant phase for most stars, because remember, the solar core is already almost hot enough for CNO cycle fusion to dominate, so it wouldn't take much contraction and heating to get CNO to take over. But the basic story given above is correct-- as the core contracts, the shell gets hotter, fusion in the shell goes berserk, and heat is deposited in the envelope, expanding it. The expansion serves to take weight off the fusing shell, lowering the amount of mass in the shell and turning down the amount of fusion, thus maintaining equilibrium with what light can diffuse out through the shell. (Note that the luminosity ends up rising even as the shell gets less and less mass in it, because its temperature is going up, and the rate that light can diffuse out also goes up as the amount of mass in the shell drops, so it is always in equilibrium as the helium ash builds up in the core).

## Ch 20 Homework

A star is in hydrostatic equilibrium when the outward push of pressure due to core burning is exactly in balance with the inward pull of gravity. When the hydrogen in a star's core has been used up, burning ceases, and gravity and pressure are no longer in balance. This causes the star to undergo significant changes. Which of the following evolutionary changes would bring a star back into hydrostatic equilibrium?

When a star's core hydrogen has been fully depleted via hydrogen burning, the star becomes unstable. The internal structure of the star changes as a result of the new instabilities within its interior. Which of the diagrams below shows the internal structure of a star immediately after running out of its core hydrogen?

Depending on its mass, it can take millions to trillions of years for a star to evolve from a main-sequence star to a red giant. Despite this astronomical length of time, astronomers are confident in their models of stellar evolution. Which of the following statements best describe why astronomers firmly believe that their models of stellar evolution are correct?

Now plug in to d=m/v, volume of sphere = 4/3 (pi) r^3, and convert 18,000 km to m

## 2 DATA ANALYSIS

### 2.1 Overview of observables

We perform an in-depth analysis of HR 7322 using interferometry (Section 2.2), asteroseismology (Section 2.3), spectroscopy (Section 2.4), and grid-based stellar modelling (Section 3.3). A graphic overview of the relationships between the variables and observational methods can be seen in Fig. 1. Starting in the right-hand side of Fig. 1, a literature value for the effective temperature determined from spectroscopy Teff,spec is used to compute the first iteration of the linear limb-darkening coefficient uλ. Combining the limb-darkening coefficient and the interferometric data, the limb-darkened angular diameter θLD of HR 7322 is found. Using a measured parallax ϖ a distance can be derived and thus the linear radius of the star Rint is then determined from equation ( 5). Finally, from θLD and the bolometric flux of the star Fbol, an estimate of the effective temperature Teff,int can be determined from interferometry using equation ( 6).

Flow diagram showing the relationships between the methods used and the derived stellar parameters.

Flow diagram showing the relationships between the methods used and the derived stellar parameters.

We wish to compare the interferometric radius of the star with that predicted from asteroseismic inference. Using photometric data from Kepler, the large frequency separation Δν and the frequency of maximum power νmax can be computed. The logarithmic surface gravity log g can be estimated from equation ( 1) using νmax and Teff,int. By anchoring log g in the spectroscopic analysis to this value, the metallicity [Fe/H] and a spectroscopic estimate of the effective temperature Teff,spec can be determined. Then Teff,spec can be fed back into a recalculation of the limb-darkening coefficient and the interferometric limb-darkened angular diameter θLD. This calculation loop continues until no change in limb-darkening coefficient is found and consequently the calculated angular diameter remains unchanged from the last iteration. An asteroseismic radius Rseismic and an asteroseismic mass Mseismic can be determined by combining asteroseismic parameters Δν and νmax with an estimate of temperature (equations 3 and 4). Finally, we compare the measured physical parameters to the quantities from stellar modelling.

### 2.2 Interferometry

We measured the angular diameter of HR 7322 using long-baseline optical interferometry. We used the PAVO beam combiner (Precision Astronomical Visible Observations Ireland et al. 2008) at the CHARA array located at Mount Wilson Observatory, California (Center for High Angular Resolution Astronomy ten Brummelaar et al. 2005). The CHARA array consists of six 1-m telescopes in a Y-configuration, allowing 15 different baseline configurations between 34.07 and 330.66 m. PAVO is a three-beam pupil-plane beam combiner, optimized for high sensitivity at visible wavelengths (∼600– |$<900>, mathrm$|⁠ ).

Our observations were made using PAVO in two-telescope mode and baselines ranging from 157.27 to 251.34 m. A summary of our observations can be found in Table 1. Table 2 lists the six stars we used to calibrate the fringe visibilities of HR 7322. Ideally an interferometric calibrator star is an unresolved point source with no close companions. The calibrator stars need to be observed as closely in time and in angular distance to the target object as possible in order to avoid changes in system variability, and therefore we observed the calibrator stars immediately before and after the target object. For all but one scan, the observing procedure was Calibrator 1 → Target → Calibrator 2. For the last scan of 2014 August 18, only one calibrator was used as the second calibrator HD 185872 caused a miscalibration of target. This does not change the derived angular diameters.

Overview of PAVO interferometric observations.

UT date . Calibrator a . Baseline b . No. of scans .
2013 July 8 acde E2W1 4
2013 July 9 ace S1W2 4
2014 Apr 8 bf E1W2 1
2014 Aug 16 cf S2E2 5
2014 Aug 17 cf E2W1 2
2014 Aug 18 cf cE2W2 3
UT date . Calibrator a . Baseline b . No. of scans .
2013 July 8 acde E2W1 4
2013 July 9 ace S1W2 4
2014 Apr 8 bf E1W2 1
2014 Aug 16 cf S2E2 5
2014 Aug 17 cf E2W1 2
2014 Aug 18 cf cE2W2 3

b The baselines have the following lengths: E2W2: 156.27 m S1W2: 210.97 m E1W2: 221.82 m S2E2: 248.13 m E2W1: 251.34 m.

c The last scan was calibrated using only c.

Overview of PAVO interferometric observations.

UT date . Calibrator a . Baseline b . No. of scans .
2013 July 8 acde E2W1 4
2013 July 9 ace S1W2 4
2014 Apr 8 bf E1W2 1
2014 Aug 16 cf S2E2 5
2014 Aug 17 cf E2W1 2
2014 Aug 18 cf cE2W2 3
UT date . Calibrator a . Baseline b . No. of scans .
2013 July 8 acde E2W1 4
2013 July 9 ace S1W2 4
2014 Apr 8 bf E1W2 1
2014 Aug 16 cf S2E2 5
2014 Aug 17 cf E2W1 2
2014 Aug 18 cf cE2W2 3

b The baselines have the following lengths: E2W2: 156.27 m S1W2: 210.97 m E1W2: 221.82 m S2E2: 248.13 m E2W1: 251.34 m.

c The last scan was calibrated using only c.

Calibrators used for the interferometric measurements. The uniform-disc angular diameter in the R band is denoted θUD,R.

HD . Sp. Type . V . K . E(BV) . θUD,R . ID .
176131 A2 V 7.08200 6.74800 0.0068 0.154 a
176626 A2 V 6.85200 6.77100 0.0219 0.147 b
177003 B2.5 IV 5.37700 5.89500 0.0145 0.204 c
179095 B8 IV 6.91500 6.99000 0.0176 0.130 d
183142 B8 V 7.06900 7.53400 0.0272 0.096 e
185872 B9 III 5.39900 5.48000 0.0252 0.266 f
HD . Sp. Type . V . K . E(BV) . θUD,R . ID .
176131 A2 V 7.08200 6.74800 0.0068 0.154 a
176626 A2 V 6.85200 6.77100 0.0219 0.147 b
177003 B2.5 IV 5.37700 5.89500 0.0145 0.204 c
179095 B8 IV 6.91500 6.99000 0.0176 0.130 d
183142 B8 V 7.06900 7.53400 0.0272 0.096 e
185872 B9 III 5.39900 5.48000 0.0252 0.266 f

Calibrators used for the interferometric measurements. The uniform-disc angular diameter in the R band is denoted θUD,R.

HD . Sp. Type . V . K . E(BV) . θUD,R . ID .
176131 A2 V 7.08200 6.74800 0.0068 0.154 a
176626 A2 V 6.85200 6.77100 0.0219 0.147 b
177003 B2.5 IV 5.37700 5.89500 0.0145 0.204 c
179095 B8 IV 6.91500 6.99000 0.0176 0.130 d
183142 B8 V 7.06900 7.53400 0.0272 0.096 e
185872 B9 III 5.39900 5.48000 0.0252 0.266 f
HD . Sp. Type . V . K . E(BV) . θUD,R . ID .
176131 A2 V 7.08200 6.74800 0.0068 0.154 a
176626 A2 V 6.85200 6.77100 0.0219 0.147 b
177003 B2.5 IV 5.37700 5.89500 0.0145 0.204 c
179095 B8 IV 6.91500 6.99000 0.0176 0.130 d
183142 B8 V 7.06900 7.53400 0.0272 0.096 e
185872 B9 III 5.39900 5.48000 0.0252 0.266 f

The angular diameters of the calibrators were found using the (VK) surface brightness calibration of Boyajian, van Belle & von Braun ( 2014). The V-band magnitudes were adopted from the Tycho-2 catalogue (Høg et al. 2000), and converted into the Johnson system using the calibration given by Bessell ( 2000). K-band magnitudes were taken from the Two Micron All Sky Survey catalogue (Skrutskie et al. 2006). Interstellar extinction was estimated from the dust map of Green et al. ( 2015) and the extinction law of O’Donnell ( 1994). The calculated angular diameters were corrected for the limb-darkening to determine the corresponding uniform-disc diameter in R band.

The data were reduced, calibrated, and analysed using the PAVO reduction pipeline, (see e.g. Ireland et al. 2008 Bazot et al. 2011 Derekas et al. 2011 Huber et al. 2012 Maestro et al. 2013). The uncertainties were estimated by performing Monte Carlo simulations with 100 000 iterations assuming Gaussian uncertainties in the visibility measurements, |$<5>, mathrm$| in the wavelength calibration, and |$<5>, mathrm$| in the sizes of the calibrator stars.

The limb-darkening coefficient uλ of HR 7322 was estimated using a Teffuλ relation in the R band (White et al., in preparation). The limb-darkening coefficient also has a metallicity and surface gravity dependence, but no strong relations with these quantities at these wavelengths were found and therefore our estimate of limb-darkening coefficient was found using only effective temperature. The relation was found by performing 10 000 iterations of a Monte Carlo simulation of the measured limb-darkening coefficients and temperatures from PAVO of 16 stars by allowing the values to vary within their uncertainties. The Sun was also added to the determination of the relation by using the limb-darkening coefficient from Neckel & Labs ( 1994). Using the spectroscopic temperature (see Table 5), the limb-darkening coefficient for HR 7322 was determined to be uλ = 0.22 ± 0.05. Using this uλ, the fit in equation ( 7) to the visibility measurements yields a limb-darkened angular diameter of HR 7322 of |$heta _> =<0.443 pm 0.007>, mathrm$| (see Fig. 2). When a uniform disc model, i.e. a model that does not include limb darkening, is fitted to the data, then the uniform-disc angular diameter is found to be |$heta _>=<0.435 pm 0.005>, mathrm$|⁠ .

Interferometric measurements of HR 7322 from PAVO. The black dots with grey error bars show the squared fringe visibility measurements, while the blue curve shows the best-fitting limb-darkened disc model. The residuals weighted by the visibility uncertainties are shown in the bottom plot.

Interferometric measurements of HR 7322 from PAVO. The black dots with grey error bars show the squared fringe visibility measurements, while the blue curve shows the best-fitting limb-darkened disc model. The residuals weighted by the visibility uncertainties are shown in the bottom plot.

An interferometric measure of effective temperature Teff,int can be found using an estimate of the bolometric flux at Earth. The bolometric flux of HR 7322 was measured by Casagrande et al. ( 2011) to be |$F_> = (1.06 pm 0.05) imes 10^<-7>, mathrm, cm>^<-2>$|⁠ , resulting in an effective temperature of |$T_> = <6350 pm 90>, mathrm$|⁠ .

### 2.3 Asteroseismology

The photometric time series of HR 7322 is available from the NASA Kepler mission, which observed HR 7322 in short-cadence mode ( ⁠|$1, mathrm$|⁠ ) during quarter 15 (Q15) spanning |$<96.7>, mathrm$| as part of Kepler Guest Investigator Programme GO40009. One safe mode event occurred during Q15, causing a gap in the photometric time series. Light curves were constructed from pixel data downloaded from the KASOC data base. 1 The raw time series was corrected for instrumental signals using the KASOC filter, which employs two median filters of different widths, with the final filter being a weighted sum of the two filters based on the variability in the light curve (Handberg & Lund 2014).

As seen in Fig. 3, the time series of HR 7322 shows a substantial number of outliers below the average flux level. As the data were obtained a few months before the second reaction wheel of the spacecraft failed, we follow the same approach as Johnson et al. ( 2014) and ascribe these outliers to pointing jitter caused by the increased friction that eventually led to the reaction wheel failure. The power density spectrum (PDS Fig. 4) used for further seismic analysis was constructed from a weighted least-squares sine-wave fitting, single-side calibrated, normalized according to Parseval’s theorem, and converted to power density by multiplying by the effective observing length obtained from the integral of the spectral window (Kjeldsen & Frandsen 1992).

Q15 short-cadence time series of HR 7322 from Kepler shown as blue points. Grey crosses show points ascribed to pointing jitter. For clarity, only 10 per cent of the data are shown.

Q15 short-cadence time series of HR 7322 from Kepler shown as blue points. Grey crosses show points ascribed to pointing jitter. For clarity, only 10 per cent of the data are shown.

Power density spectrum of HR 7322. The full spectrum is shown in grey with a |$<3>, mu mathrm$| Epanechnikov smoothed version overlain in black. The fitted spectrum from the peak-bagging procedure is overlain in red. The markers indicate the frequency and angular degree of the fitted modes.

Power density spectrum of HR 7322. The full spectrum is shown in grey with a |$<3>, mu mathrm$| Epanechnikov smoothed version overlain in black. The fitted spectrum from the peak-bagging procedure is overlain in red. The markers indicate the frequency and angular degree of the fitted modes.

The individual mode frequencies for HR 7322 were extracted from the power spectrum using the peak-bagging approach described in Lund et al. ( 2017). Fig. 4 shows the PDS with the frequency of the fitted modes indicated, and as seen here HR 7322 shows a departure from the regularity in the mode degree pattern around |$<780>, mu mathrm$| with two dipole modes (green triangles) being between two radial modes (orange diamonds) instead of only a single dipole mode. First guesses for the mode frequencies included in the peak-bagging were obtained from visual inspection of the PDS. We note that l = 1 modes were treated in the same way as pure p modes, but with amplitudes and linewidths decoupled from the l = 0 modes.

The large frequency separation Δν and the frequency of maximum power νmax were estimated by running the cleaned time series through the automated analysis pipeline described in Huber et al. ( 2009, 2011), and they were determined to be |$Delta u = <53.92 pm 0.20>, mu mathrm$| and |$u _ ext = <960 pm 15>, mu mathrm$|⁠ . The value of νmax is in agreement with a simple Lorentz fit to the amplitudes of the individual modes.

Takeda et al. ( 2005) measured the projected rotational velocity Vsin i of HR 7322 using spectroscopy to be 3 km s −1 , indicating either a low rotation rate or a pole-on view. From the peak-bagging, no clear independent values can be obtained for the rotational splitting (νs) or stellar inclination. However, a seismic equivalent for the projected rotational velocity can be derived from the projected rotational splitting νssin i and the modelled stellar radius as Vsin i = 2πRνssin i (Lund et al. 2014). We find a value of |$V sin i = 4.5pm 1.8, m km, s^<-1>$|⁠ , in agreement with the value from Takeda et al. ( 2005).

### 2.4 Spectroscopy

The Hertzsprung SONG 1-m telescope (Andersen et al. 2014 Grundahl et al. 2017) at Observatorio del Teide on Tenerife was used to obtain high-resolution (R = 90 000) échelle spectra of HR 7322 on 2016 March 13 and September 16. Extraction of spectra, flat fielding, and wavelength calibration were carried out with the SONG data reduction pipeline. Individual spectra were combined in iraf after correction for Doppler shifts resulting in a spectrum in the ∼4400– |$<6900>, mathrm>$| region with a signal-to-noise ratio of S/N ∼ 400 at |$<6000>, mathrm>$|⁠ . For this spectrum, equivalent widths of the spectral lines listed in Nissen ( 2015, table 2) were measured by Gaussian fitting to the line profiles.

The equivalent widths were analysed with marcs model atmospheres (Gustafsson et al. 2008) with the method described in Nissen et al. ( 2017) to obtain abundances of elements. As seen in equation ( 1), the frequency of maximum power is related to the surface gravity and the effective temperature |$u _ ext propto g / sqrt<>>>$|⁠ . A logarithmic surface gravity of log g = 3.95 ± 0.01 was determined for HR 7322 by using the asteroseismic νmax and the interferometric Teff (see Table 5) and by adopting |$T_< ext,odot > = <5772>, mathrm$|⁠ , log g = 4.438, and |$u _< ext,odot > = <3090>, mu mathrm$| for the Sun. Then, the spectroscopic Teff was determined from the requirement that the same Fe abundance should be obtained from Fe i and Fe ii lines. In this connection, non-LTE corrections from Lind, Bergemann & Asplund ( 2012) were taken into account, which decreases Teff by |$<50>, mathrm$| relative to the LTE value. The results are |$T_> = <6313 pm 35>, mathrm$| and [Fe/H] = −0.23 ± 0.04. We assume that the systematic uncertainties are of the same order of magnitude as the statistical uncertainties and add these uncertainties in quadrature to get a combined uncertainty of |$<50>, mathrm$| and |$<0.06>, mathrm$|⁠ , respectively. Comparing this effective temperature from spectroscopy with the effective temperature from interferometry, we see that they have an excellent agreement within ∼0.4σ. Using this spectroscopic Teff value in the scaling relation does not change log g significantly, i.e. by only |$<0.003>, mathrm$|⁠ . As the two temperatures agree, we choose to use the interferometric temperature in the following analysis.

In addition, the ratio between the abundance of alpha-capture elements (Mg, Si, Ca, and Ti) and Fe was determined to be [α/Fe] = 0.06 ± 0.03 showing that HR 7322 belongs to the population of low-α (thin disc) stars.

## Evolutionary Tracks

Let&rsquos now use these ideas to follow the evolution of protostars that are on their way to becoming main-sequence stars. The evolutionary tracks of newly forming stars with a range of stellar masses are shown in Figure (PageIndex<1>). These young stellar objects are not yet producing energy by nuclear reactions, but they derive energy from gravitational contraction&mdashthrough the sort of process proposed for the Sun by Helmhotz and Kelvin in this last century (see the chapter on The Sun: A Nuclear Powerhouse).

Figure (PageIndex<1>) Evolutionary Tracks for Contracting Protostars. Tracks are plotted on the H&ndashR diagram to show how stars of different masses change during the early parts of their lives. The number next to each dark point on a track is the rough number of years it takes an embryo star to reach that stage (the numbers are the result of computer models and are therefore not well known). Note that the surface temperature (K) on the horizontal axis increases toward the left. You can see that the more mass a star has, the shorter time it takes to go through each stage. Stars above the dashed line are typically still surrounded by infalling material and are hidden by it.

Initially, a protostar remains fairly cool with a very large radius and a very low density. It is transparent to infrared radiation, and the heat generated by gravitational contraction can be radiated away freely into space. Because heat builds up slowly inside the protostar, the gas pressure remains low, and the outer layers fall almost unhindered toward the center. Thus, the protostar undergoes very rapid collapse, a stage that corresponds to the roughly vertical lines at the right of Figure (PageIndex<1>). As the star shrinks, its surface area gets smaller, and so its total luminosity decreases. The rapid contraction stops only when the protostar becomes dense and opaque enough to trap the heat released by gravitational contraction.

When the star begins to retain its heat, the contraction becomes much slower, and changes inside the contracting star keep the luminosity of stars like our Sun roughly constant. The surface temperatures start to build up, and the star &ldquomoves&rdquo to the left in the H&ndashR diagram. Stars first become visible only after the stellar wind described earlier clears away the surrounding dust and gas. This can happen during the rapid-contraction phase for low-mass stars, but high-mass stars remain shrouded in dust until they end their early phase of gravitational contraction (see the dashed line in Figure (PageIndex<1>)).

To help you keep track of the various stages that stars go through in their lives, it can be useful to compare the development of a star to that of a human being. (Clearly, you will not find an exact correspondence, but thinking through the stages in human terms may help you remember some of the ideas we are trying to emphasize.) Protostars might be compared to human embryos&mdashas yet unable to sustain themselves but drawing resources from their environment as they grow. Just as the birth of a child is the moment it is called upon to produce its own energy (through eating and breathing), so astronomers say that a star is born when it is able to sustain itself through nuclear reactions (by making its own energy.)

When the star&rsquos central temperature becomes high enough (about 12 million K) to fuse hydrogen into helium, we say that the star has reached the main sequence (a concept introduced in The Stars: A Celestial Census). It is now a full-fledged star, more or less in equilibrium, and its rate of change slows dramatically. Only the gradual depletion of hydrogen as it is transformed into helium in the core slowly changes the star&rsquos properties.

The mass of a star determines exactly where it falls on the main sequence. As Figure (PageIndex<1>) shows, massive stars on the main sequence have high temperatures and high luminosities. Low-mass stars have low temperatures and low luminosities.

Objects of extremely low mass never achieve high-enough central temperatures to ignite nuclear reactions. The lower end of the main sequence stops where stars have a mass just barely great enough to sustain nuclear reactions at a sufficient rate to stop gravitational contraction. This critical mass is calculated to be about 0.075 times the mass of the Sun. As we discussed in the chapter on Analyzing Starlight, objects below this critical mass are called either brown dwarfs or planets. At the other extreme, the upper end of the main sequence terminates at the point where the energy radiated by the newly forming massive star becomes so great that it halts the accretion of additional matter. The upper limit of stellar mass is between 100 and 200 solar masses.

## Pourquoi la trace de l'étage sous-géant est-elle presque horizontale sur le diagramme HR?

Après la séquence principale, lorsque la fusion s'affaiblit ou s'arrête dans le cœur, le rayonnement extérieur s'affaiblit. Le noyau d'hélium se contracte et s'échauffe. L'énergie gravitationnelle est à nouveau convertie en énergie thermique!

L'étoile semblera refroidir lentement et subira une légère augmentation de luminosité. Au cours de cette phase, le chemin que suivra l'étoile dans le diagramme HR est presque horizontal à droite de sa position sur la séquence principale. Les étoiles de cette phase sont généralement appelées sous-géantes.

Mais je ne comprends toujours pas.

Je remplacerai la réponse précédente pour me concentrer sur la branche "sous-géante" avant la géante rouge, plutôt que sur la séquence pré-principale ou la "branche horizontale" de la fusion centrale de l'hélium. Ce sont d'autres fois où la luminosité est constante, mais cette question concerne la branche sous-géante, que j'ai manquée auparavant.

La raison pour laquelle la luminosité est presque constante sur la branche sous-géante est liée à la "relation masse-luminosité" des étoiles de la pré-séquence et de la séquence principale. Elle est due à la diffusion radiative et à la manière dont elle conduit à une luminosité qui ne dépend que de la masse, pour une composition donnée. Si vous vous comparez à des pistes pré-séquence principale, vous devriez constater que les sous-géants retracent plus ou moins cette évolution antérieure, juste avec une luminosité un peu plus élevée parce que de nombreux électrons ont été avalés dans des neutrons, réduisant l'opacité et augmentant le tout. taux important de diffusion radiative. C'est essentiellement une relation masse-luminosité dominée par l'hélium, au lieu d'être dominée par l'hydrogène, car le rayon augmente en raison des détails de la façon dont l'intérieur évolue.

La raison pour laquelle la luminosité augmente finalement fortement sur la branche géante rouge est que lorsque le noyau dégénéré commence à construire en masse, il commence à contrôler la température de la région de fusion, et cela modifie considérablement la structure interne de manière à expliquer les géantes rouges .

Cela se produit en raison de l'épuisement de H au cœur:

Épuisement de H au cœur -> contraction du cœur -> augmentation de la température du cœur + démarrage de la fusion H dans l'enveloppe (dans le diagramme HR, l'étoile monte) -> pour maintenir l'équilibre de l'injection d'énergie de fusion H dans l'enveloppe, l'enveloppe se dilate et se refroidit (se déplaçant horizontalement vers la droite dans le diagramme HR) -> .

## Why is the track of the subgiant stage almost horizontal on the HR diagram? - Astronomy

Star clusters provide excellent test sites for the theory of stellar evolution. Every star in a given cluster formed at the same time, from the same interstellar cloud, with virtually the same composition. Only the mass varies from one star to another. This allows us to check the accuracy of our theoretical models in a very straightforward way. Having studied in some detail the evolutionary tracks of individual stars, let us now consider how their collective appearance changes in time.

In Chapter 17 we saw how astronomers estimate the ages of star clusters by determining which of their stars have already left the main sequence. (Sec. 17.10) In fact, the main-sequence lifetimes that go into those age measurements represent only a tiny fraction of the data obtained from theoretical models of stellar evolution. Starting from the zero-age main sequence, astronomers can predict exactly how a newborn cluster should look at any subsequent time. Although we cannot see into the interiors of stars to test our models, we can compare stars' outward appearances with theoretical predictions. The agreement—in detail—between theory and observation is remarkably good.

We begin our study shortly after the cluster's formation, with the upper main sequence already fully formed and burning steadily, and lower-mass stars just beginning to arrive on the main sequence, as shown in Figure 20.16(a). The appearance of the cluster at this early stage is dominated by its most massive stars—the bright blue supergiants. Now let's follow the cluster forward in time and see how its H—R diagram evolves.

Figure 20.16 The changing H—R diagram of a hypothetical star cluster. (a) Initially, stars on the upper main sequence are already burning steadily while the lower main sequence is still forming. (b) At 10 7 years, O-type stars have already left the main sequence, and a few red giants are visible. (c) By 10 8 years, stars of spectral type B have evolved off the main sequence. More red giants are visible, and the lower main sequence is almost fully formed. (d) At 10 9 years, the main sequence is cut off at about spectral type A. The subgiant and red-giant branches are just becoming evident, and the formation of the lower main sequence is complete. A few white dwarfs may be present. (e) At 10 10 years, only stars less massive than the Sun still remain on the main sequence. The cluster's subgiant, red-giant, horizontal, and asymptotic giant branches are all discernible. Many white dwarfs have now formed.

Figure 20.16(b) shows the appearance of our cluster's H—R diagram after 10 million years. The most massive O-type stars have evolved off the main sequence. Most have already exploded and vanished, as just discussed, but one or two may still be visible as red giants. The remaining cluster stars are largely unchanged in appearance—their evolution is slow enough that little happens to them in such a relatively short period of time. The cluster's H—R diagram shows the main sequence slightly cut off, along with a rather poorly defined red-giant region. Figure 20.17 shows the twin open clusters h and (the Greek letter chi) Persei, along with their combined H—R diagram. Comparing Figure 20.17(b) with such diagrams as those in Figure 20.16, astronomers estimate the age of this pair of clusters to be about 10 million years.

Figure 20.17 (a) The "double cluster" h and Persei. (b) The H—R diagram of the pair indicates that the stars are very young—probably only about 10 million years old.

After 100 million years (Figure 20.16c) stars brighter than type B5 or so (about 4׫ solar masses) have left the main sequence, and a few more red supergiants are visible. By this time most of the cluster's low-mass stars have finally arrived on the main sequence, although the dimmest M stars may still be in their contraction phase. The appearance of the cluster is now dominated by bright B stars and brighter red giants.

At any time during the evolution the cluster's original main sequence is intact up to some well-defined stellar mass, corresponding to the stars that are just leaving the main sequence at that instant. We can imagine the main sequence being "peeled away" from the top down, with fainter and fainter stars turning off and heading for the giant branch as time goes on. Astronomers refer to the high-luminosity end of the observed main sequence as the main-sequence turnoff. The mass of the star that is just evolving off the main sequence at any moment is known as the turnoff mass.

At 1 billion years, the main-sequence turnoff mass is around 2 solar masses, corresponding roughly to spectral type A2. The subgiant and giant branches associated with the evolution of low-mass stars are just becoming visible, as indicated in Figure 20.16(d). The formation of the lower main sequence is now complete. In addition, the first white dwarfs have just appeared, although they are often too faint to be observed at the distances of most clusters. Figure 20.18 shows the Hyades open cluster and its H—R diagram. The H—R diagram appears to lie between Figures 20.16(c) and 20.16(d), suggesting that the cluster's age is about 500 million years.

Figure 20.18 (a) The Hyades cluster, a relatively young group of stars visible to the naked eye. (b) The H—R diagram for this cluster is cut off at about spectral type A, implying an age of about 500 million years.

At 10 billion years, the turnoff point has reached solar-mass stars, of spectral type G2. The subgiant and giant branches are now clearly discernible (see Figure 20.16e), and the horizontal and asymptotic giant branches appear as distinct regions in the H—R diagram. Many white dwarfs are also present in the cluster. Although stars in all these evolutionary stages are also present in the 1-billion-year-old cluster shown in Figure 20.16(d), they are few in number—typically only a few percent of the total number of stars in the cluster. Also, because they evolve so rapidly, they spend very little time in these regions. Low-mass stars are much more numerous and evolve more slowly, so their evolutionary tracks are more easily detected.

Figure 20.19 shows the globular cluster 47 Tucanae. By carefully adjusting their theoretical models until the cluster's main sequence, subgiant, red-giant, and horizontal branches are all well matched, astronomers have determined its age to be roughly 11 billion years, a little older than our hypothetical cluster in Figure 20.16(e). In fact, globular cluster ages determined in this way show a remarkably small spread. All the globular clusters in our Galaxy appear to have formed between about 10 and 12 billion years ago.

Figure 20.19 (a) The southern globular cluster 47 Tucanae. (b) Fitting its main-sequence turnoff and its giant and horizontal branches to theoretical models gives 47 Tucanae an age of about 11 billion years, making it one of the oldest known objects in the Milky Way Galaxy. The inset is a high-resolution ultraviolet image of 47 Tucanae's core region, taken with the Hubble Space Telescope and showing many "blue stragglers"—massive stars lying on the main sequence above the turnoff point, resulting perhaps from the merging of binary star systems (see also Figure 20.9). The points representing white dwarfs, red dwarfs, and blue stragglers are for illustration only—they have been added to the original data set, based on Hubble observations of this and other clusters.

Recent Hubble observations of nearby globular clusters have also revealed for the first time the white dwarf sequences long predicted by theory but previously too faint to detect at such large distances. Figure 20.20 (b) shows a cluster called M4 on image of white dwarfs in HST lying 2100 pc from Earth. To illustrate our point, we have combined in Figure 20.19(b) both the newer white-dwarf data for M4 and the older main-sequence/red-giant data for 47 Tuc. The complete H-R diagram for a single cluster is expected to look qualitatively similar.

Figure 20.20 (a) The globular cluster M4, as seen through a large ground-based telescope on Kitt Peak Mountain. This is the closest globular cluster to us, at 2100 pc away it spans some 16 pc. (b) A peek at M4's suburbs by the Hubble telescope shows nearly a hundred white dwarfs within a small 0.4-pc area, some of the brightest ones are circled in blue.

Stellar evolution is one of the great success stories of astrophysics. Like all good scientific theories, it makes definite testable predictions about the universe while remaining flexible enough to incorporate new discoveries as they occur. Theory and observation have advanced hand in hand. At the start of the twentieth century many scientists despaired of ever knowing even the compositions of the stars, let alone why they shine and how they change. Today, the theory of stellar evolution is a cornerstone of modern astronomy.

## Lecture 16 - Stellar Evolution (3/16/99)

The Ring Nebula M57, courtesy HST/STSCI

Planetary nebulae are the outer layers of a star, blown away during the thermal pulsing of the helium shell-burning phase of its evolution. The Hubble Space telescope has snagged a number of spectacular images of planetary nebulae, showing the variety of rings blown by these dying stars.

Mass loss is a general phenomenon, with both AGB stars and hot main sequence stars losing significant amounts of their envelopes during various stages in their lives.

• The Ring Nebula M57
• The Southern Ring Nebula NGC3132
• Gas being ejected in stellar wind by hot star WR124
• A Turtle Nebula, NGC 6210?
• Young planetary nebula the Stingray
• Hubble captures the shrouds of dying stars Witness final blaze of Sun-like stars
• The doomed star Eta Carinae
• The planetary nebula NGC 7027
• The spectacular Hourglass Nebula!

## Stellar Evolution in Outline:

1. The Life Cycles of Stars
• Stars have "lives" in that they are born out of dust and gas, grow under gravity, start burning nuclear fuel and become full-fledged stars, go through stages as different fuel sources are found, exhaust their energy and die.
• Stars return much (in fact sometimes most) of their mass back to the interstellar medium, from which new generations of stars are born.
• The Sun is 4.5 billion years old, while the oldest stars known appear to be 18 billion years old. The Sun is not a first generation star, it was formed from material that had been processed through at least one generation of stars that lived and died previously!
• Almost all the elements heavier than helium were "created" from fusion in the cores of stars. The material that our bodies are made of was synthesized in the center of stars now long dead!
• The understanding of the stages of a star's life comes from understanding the Hertzsprung-Russell (H-R) diagram of L vs. T.
• A star like the Sun spends 90% of its life burning hydrogen on the main sequence. Eventually the hydrogen runs out and the star must change its ways. This is when things get interesting.
2. The Tale of the Core
• When hydrogen is exhausted in the center (only about the innermost 10% of the stars mass is available for fusion as we will see), the core, which is now made of mostly helium, must contract to derive gravitational energy to keep up the pressure against gravity.
• This contraction is slow, as the Sun could actually generate its current luminosity by contraction of 40 meters per year! Q: How long can the Sun keep this up before contracting to nothing?
• As the helium core contracts, it heats up, and this ignites the unfused hydrogen in a shell around the core. This further liberates new energy which flows out into the envelope of the star.
• The envelope structure was set up to be in equilibrium with the luminosity of the hydrogen burning core. Now it has extra energy pumped in at the bottom, and it is too dense to let all this new heat and radiation pass through. To compensate it will expand in radius, dropping in density, and thus in surface temperature at the outside.
• The radius will increase greatly, by as much as 25 times! The star moves to the right (becomes cooler) in the H-R diagram.
• Eventually the envelope will drop in density enough to allow the new luminosity to come out through the surface, and the star will increase in luminosity, moving upward on the H-R diagram.
• Thus, after hydrogen core burning on the main sequence, a star move upward and to the right (higher L, lower T).
• The star is much larger in radius, and is called a giant star. It is cooler, and thus redder in color, so it is commonly called a red giant.
• Some stars are even larger than these giants, with radii almost a thousand times the radius of the Sun. These are called supergiants.
• Supergiants are high mass stars that have evolved off the main sequence. Betelgeuse in the constellation Orion is a red supergiant with mass 15 Msun and radius of nearly 1000 Rsun (almost 4 AU!).
• Blue giants and supergiants usually refer to high-mass (and thus high temperature) main sequence stars.
3. Low Mass Stars
• Stars less than 1/12 solar mass are called brown dwarfs. The central temperature of these stars never gets high enough to fuse hydrogen. These are "failed stars" and are very faint. They can be thought of as large "Jupiters".
• There may be very many brown dwarfs in our galaxy, though we wouldn't see them easily. Recently, the first brown dwarfs were discovered in binary systems, where their masses could be deduced.
• Stars between 1/12 and 0.4 solar masses are called red dwarfs. These are main sequence hydrogen burning stars, but cooler, smaller, and fainter than the Sun (hence "red" and "dwarf").
• Red dwarfs are much less luminous than the Sun. The approximate mass-luminosity relation (L/Lsun) = (M/Msun)^3.5 leads to the lifetime (t/tsun) = (M/L)/(Msun/Lsun) = (Msun/M)^2.5, so a star with mass 0.1 Msun would have a lifetime of 300 times the Sun's lifetime of 10 billion years, or 300 billion years! Red dwarf stars burn their fuel so slowly that they live a very long time (the ultimate conservationists).
• Red dwarf stars, because of their low luminosities, are fully convective - that is they transport heat from their cores to the surface by gas currents. This also mixes the stars fully so that they can use all of the hydrogen in the star to burn, and the helium made in the core will be mixed throughout the star.
• When a red dwarf has used up all its hydrogen, it will simply collapse under gravity until it becomes a helium white dwarf (see below). Then it will fade away fainter and cooler until it is a cold dead ball of gas.
4. Medium Mass Stars
• Stars from about 0.4 to 3 solar masses are normal hydrogen fusing main sequence stars like our Sun for most of their lives.
• Stars with masses less than about 1.1 solar masses have no convection in their cores. Heat transport is purely radiative out into the envelope. Most importantly, there is therefore no mixing of unburnt envelope hydrogen into the core. Once the core uses up its fuel, it has no recourse.
• Only about the inner 10% to 13% of the mass of the star is available for fusion in the hot core. This is why the lifetime of the Sun on the main sequence is 10 billion years, not 80 billion years. Gas outside the core is not available for fuel.
• Stars from 1.1 to 3 solar masses have some convection in their core region. However, this can only mix in around a few percent of the mass near the core, so this does not make much difference in the lifetimes.
• In all these cases, when the core hydrogen has been fused to helium, we are left with an inert helium core.
• The helium core cannot maintain pressure against the weight of the star above it, and to generate heat and thus pressure it contracts releasing gravitational energy. This is slow contraction by about 40 meters per year to maintain 1 solar luminosity of energy.
• The increase in temperature from the gravitational contraction of the core raises the temperature of the hydrogen containing layers just outside the core, causing fusion to occur. At this point hydrogen fusion is going on in a shell around the contracting core. This stage is called hydrogen shell burning.
• The energy from the core contraction and the shell burning flows out into the envelope. The structure of the envelope was set up to deal with the main sequence luminosity which was approximately constant. Now it cannot handle this extra luminosity so it expands outward becoming larger in radius, less dense, and thus lower temperature ( T^4 proportional to L/R^2 with R increasing).
• At first only a little of the extra luminosity gets out, with most of the energy going into expanding the envelope. Eventually the stare grows in size by around 25 times its initial radius, and it becomes low enough density so that the radiation can flow out at the rate it is being generated. The luminosity now has grown by a factor of 10 or so.
• The star is now cooler ("redder") and larger ("a giant"), and is called a red giant.
• Red giants are located to the right and up from the main sequence. They form a locus of stars called the giant branch.
• On the giant branch, the helium core is contracting and getting hotter and hotter. These stars are massive enough that the core eventually gets hot enough (100 million K) to fuse helium into carbon (and nitrogen and oxygen) in the triple-alpha reaction.
• How this fusion begins depends upon the state of the core at this time. Stars with masses below about 2 solar masses have cores that are degenerate (see below) when helium fusion begins. This means that the helium fusion begins all at once in a big flash called the helium flash. In about 1 second the core emits the equivalent luminosity of 10 billion suns! You dont notice anything from the outside because this energy is absorbed by the core.
• Stars with more the 2 solar masses do not have degenerate cores at the time of helium fusion, so they start fusion gradually.
• In any event, helium fusion in the core stops the contraction, and expands the core slightly, and the envelope contracts a little increasing the temperature. The star moves to the left a little and slightly down on the HR diagram after reaching the top of the giant track at the helium flash. This is called the horizontal branch, since it extends to the left horizontally from the giant branch.
• The helium is burning in the core and hydrogen is burning in the shell around the core for stars on the horizontal branch.
• After the helium is done burning in the core, the core, which is now made of carbon, nitrogen and oxygen, contracts again heating up and starting helium burning in a shell (with a hyrogen shell further out).
• Thus, with helium and hydrogen burning in concentric shells, the star moves up the giant branch again, becoming even larger. This track parallels but is slightly to the left (hotter) than the giant branch, and is called the asymptotic giant branch. This branch extends a factor of 10 to 100 times the luminosity of the giant branch, and stars on this branch are called red supergiants.
• For these medium mass stars with less than 3 Msun, the core never gets hot enough for the carbon to fuse. The helium core contracts until it becomes degenerate.
5. Degenerate Matter
• We have talked about degenerate matter as if it is some special state that stuff turns into at high density. What is it?
• Upon collapse, the helium and C-N-O cores collapse to about 0.01 of their initial radius, thus increasing the density by a factor of 10^6.
• The electrons and nuclei of the ionized gas are squeezed together tighter and tighter. Do you remember what happened when we tried to confine an electron into a small orbit around the nucleus?
• We found that we had to deal with the wave nature of the electron. Interference effects allowed us to have only integer numbers of waves in the orbit -> energy levels.
• The requirement that we be able to distinguish electron led us to allowing only one electron of a given spin (either spin "up" or spin "down") in each wavelength (orbital).
• The same thing happens here. If you confine an electron into a radius r, the smallest you can make this is 2pi r = L where L is the wavelength: L = h/ mv.
• Thus, we are limited to confining the electron into a space

thus for a given velocity, the electron can be confined to a space no smaller than

## Watch the video: Spectral Classes, Luminosity, and the H-R Diagram (July 2022).

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