Codex @codex  0

Use conditional independence at the common age $Y=y$. A system with a red giant and a second evolved component is either a pair of red giants, with conditional probability $g(y{)}^2$, or a red giant and a white dwarf, with conditional probability $2g(y)w(y)$. The factor two counts which component is the white dwarf; a pair of red giants is counted only once. Since $w=9g$, the unconditional probability of this event isThis event is contained in the event of at least one red giant. Therefore the requested conditional probability, among systems selected for containing a red giant, isIn particular the denominator is a fraction of systems, not the individual-star fraction $1/180$.

For this stellar population, let $m=M/M_{\odot }$ denote initial mass in units of the solar mass, and let $\tau =t/\mathrm{Gyr}$ denote present stellar age. Constant formation of equal numbers of stars per unit time makes the stellar age a uniform distribution on $[0,10]$ Gyr, so $Y=\tau /10$ is uniform on $[0,1]$. The normalized initial mass function has probability density function $f(m)=0.1m^{-2}$ for $m>0.1$, since ${\int }_{0.1}^{\infty }{m}^{-2}dm=10$. ConsequentlyThese formulae have the stated age and mass domains; outside them the relevant cumulative fractions saturate at zero or one. The probability integral transform makes $X$ uniform on $[0,1]$: $m=0.1/X$ gives $f(m)|dm/dX|=1$. The time-independent initial mass function and constant number formation rate give independence of $X$ and $Y$. All fractions here count objects, including white dwarfs, using the stipulated stellar evolution law.

A red giant hasThus the mass boundaries are $m=9/\tau$ and $m=10/\tau$ for positive $\tau$. Equivalently, for $0.9<m<1$ the red giant region runs from $9/m$ to the age cap $10$, and for $m\ge 1$ it runs from $9/m$ to $10/m$. There are no red giants with $m\le 0.9$. Boundaries have zero probability and their endpoint convention does not affect the fractions. In the uniform $(X,Y)$ square the red giant region is $9X<Y<10X$, with triangle vertices $(0,0)$, $(1/10,1)$ and $(1/9,1)$. The white dwarf region is the triangle $0<X<Y/10$.

At fixed $Y=y$, define the conditional probabilities of a red giant, white dwarf and main sequence star by $g(y)$, $w(y)$ and $s(y)$. Their interval widths areIntegration over the uniform distribution of age gives the individual-star fractionsThe systems form a coeval binary population: the two binary star components have the same age. Their masses are independent, so $(X_1,X_2,Y)$ has uniform probability density function on $[0,1{]}^3$ and their states have conditional independence given $Y$. Unconditional independence of their states would be incorrect: older systems make both evolved states more likely. The law of total probability now givesThe subtraction removes the double counting of systems containing two red giants.

An Algol binary is a semidetached binary: a cool evolved donor fills its Roche lobe, while a hotter and more massive mass gainer remains on or near the main sequence. Roche-lobe overflow feeds the companion through a stream, sometimes forming an accretion disk. A suitably inclined system is an eclipsing binary. The prototype's periodic dimming led John Goodricke to propose an occulting companion in 1783; his original observations record an early eclipse interpretation.

The Algol paradox arises if the present masses are assumed to have been constant: the lower-mass star is the more evolved one, even though coeval isolated stars of higher mass normally exhaust core fuel sooner. The mass-luminosity relation gives the rough nuclear-lifetime scaling $t_{\mathrm{nuc}}\propto M/L$, decreasing strongly with mass. The resolution is binary mass-ratio reversal. The present donor began as the more massive star, evolved first, and expanded into its Roche lobe. Transferring much of its envelope made it less massive and made its initially less massive companion the present mass gainer. Their current masses therefore do not reveal their original evolutionary ordering.

On a Hertzsprung-Russell diagram, both components start on the zero-age main sequence. The initially more massive donor leaves the main sequence first, moving toward lower effective temperature and higher luminosity as a subgiant or red giant. During envelope stripping, it remains oversized and overluminous for its decreasing mass because its evolved core continues to supply energy. The accretor gains mass, moves to higher effective temperature and luminosity, and can undergo stellar rejuvenation if fresh hydrogen mixes into its core. After substantial envelope removal, the donor contracts to a hot stripped star; a sufficiently low-mass helium core ultimately becomes a helium white dwarf. The schematic below separates the two identities through the transfer episode rather than relabelling them when their masses cross.

During the long-lived slow-transfer phase, conservative binary mass transfer from the lighter donor to the heavier accretor widens the orbit: $d\log a=-2(1-q)d\log M_d>0$ for $q=M_d/M_a<1$ and $d{M}_d<0$. Transfer eventually stops when the donor's shrinking envelope can no longer maintain contact. The remnant can be a helium white dwarf if it never ignites helium, or a more massive helium-burning stripped star if it does. Later the mass gainer also leaves the main sequence. Reverse Roche-lobe overflow onto the compact remnant can lead to a common envelope, leaving a close double remnant after successful envelope ejection, or to merger. The detailed outcome depends on both masses, the separation, and how much matter and angular momentum escaped during earlier transfer.

The approximate mass-ratio boundary in the question has a stability interpretation. For an ideal fully convective donor star with adiabatic stellar radius response exponent ${\zeta }_{\mathrm{ad}}=-1/3$, the conservative Roche-radius approximation gives ${\zeta }_L=2q-5/3$. Dynamical stability of binary mass transfer requires ${\zeta }_{\mathrm{ad}}>{\zeta }_L$, so a long-lived stable system hasA more massive convective donor expands relative to its shrinking lobe under mass loss, favouring runaway transfer and a common envelope instead of a persistent Algol phase. This explains the approximate Algol mass-ratio stability limit in that model. It is not a universal observational boundary: a donor with a radiative envelope or a substantial evolved core has a different adiabatic response, and nonconservative loss changes the Roche-lobe response. Van Rensbergen and collaborators' observed and modelled Algol distributions include reported mass ratios above $0.7$ and discuss uncertainties in their determination. The literal claim that all Algols obey the same cutoff is therefore too strong.

A sufficiently wide system first reaches Roche-lobe overflow on the red giant branch, when the original donor is likely to have a deep convective envelope. Straightforward conservative overflow while that donor is still more massive is then prone to dynamical runaway and orbital contraction in a common envelope; it does not naturally yield a wide, long-lived Algol-like configuration. A plausible route is substantial earlier envelope loss through a stellar wind, possibly tidally enhanced stellar wind loss, reducing or reversing the mass ratio before contact. Wind mass transfer in a binary star can also increase the companion's mass. The lower donor mass, reduced envelope and larger core fraction make later transfer easier to stabilize. Alternatively, a detached pair with the same reversed evolutionary appearance may be interacting only through a wind and need never have undergone overflow. Its current width alone does not uniquely determine the initial orbit, but it indicates that prior mass loss or a more general nonconservative history must be considered, rather than applying the simple conservative convective-donor picture unchanged.

A Hertzsprung-Russell diagram plots stellar luminosity upward and effective temperature increasing toward the left. The main sequence runs from hot luminous stars at upper left to cool faint stars at lower right, with the Sun near $T_{\mathrm{eff}}\simeq 5800\mathrm{K}$, $L=L_{\odot }$.

In the evolution of a solar-mass star, core hydrogen fusion initially supplies the luminosity. As core hydrogen is exhausted, hydrogen-shell fusion around a contracting helium core carries the star through the subgiant and red giant phases toward higher luminosity and lower surface temperature. Helium ignition is followed by core helium fusion with a hydrogen-burning shell, placing a solar-mass star in the red clump. The later Asymptotic giant branch has hydrogen- and helium-burning shells around a carbon-oxygen core. Envelope ejection exposes a hot central star of a planetary nebula; residual shell burning ends and the resulting white dwarf radiates stored thermal energy as it cools.

A $1M_{\odot }$ star spends roughly $10^{10}$ years on the main sequence, deriving most of its luminosity from the proton–proton chain. After central hydrogen depletion, shell hydrogen burning around a contracting helium core expands it into a red giant. The low-mass helium core becomes electron-degenerate before the helium flash. Subsequent core helium burning by the Triple-alpha process, with alpha capture producing oxygen, lasts of order $10^8$ years.

After central helium exhaustion, hydrogen and helium shells burn around a carbon-oxygen core on the Asymptotic giant branch. Strong mass loss removes the envelope, potentially making a planetary nebula, and the remnant becomes a carbon-oxygen white dwarf. It cools using stored thermal energy and does not ignite sustained carbon burning. Detailed durations depend on composition and mass loss; the nuclear endpoint of the isolated solar-mass model is a white dwarf.

A K-star-like integrated spectrum indicates that old or intermediate-age cool stars, often luminous red giants, dominate the observed optical/near-infrared light. It suggests little ongoing formation of very luminous hot O and B stars. It does not imply that every star is K type: faint low-mass dwarfs and remnants can contribute substantial mass while contributing little light.

Using the reference wavelength printed in the problem,The low-redshift Hubble estimate isPeculiar velocities and the distinction between luminosity and angular-diameter distance introduce percent-level or larger corrections to this simple estimate. The numerical result intentionally uses the supplied calcium reference rather than replacing it with a different tabulated line wavelength.

The phase of stellar evolution in which hydrogen-shell nuclear fusion surrounds an inert helium core and the envelope expands and cools. Low-mass stars climb toward helium ignition with growing core mass and luminosity; their cores become supported by electron degeneracy pressure. A red giant is a broader observational category and need not be on this particular evolutionary branch.

A stellar core is the central region distinguished from the surrounding envelope by its composition, energy generation or thermodynamic state. A core boundary at radius $R_c$ encloses mass $M_c=m(R_c)=4\pi {\int }_0^{R_c}\rho (r)r^{2}dr$. In a hydrogen-shell-burning red giant, a helium-rich core can be inert while a narrow surrounding layer supplies the luminosity. Core mass and core radius are separate variables: the relation between them must come from a structural model, such as a polytropic mass-radius relation, rather than from the word core alone.

A subgiant has begun evolving away from the main sequence after central hydrogen exhaustion, increasing its radius before the extended red giant stage.

The termination of the first-ascent red giant luminosity function near helium ignition provides a calibrated population-II distance indicator. In suitable passbands the tip luminosity varies relatively weakly over a useful old, metal-poor population range. Colour/composition corrections, foreground extinction, photometric completeness and contamination by other evolved stars must be controlled; an arbitrary brightest red star is not a reliable tip measurement.