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For a circular binary with negligible spin, the two orbital radii about the center of mass are $a_1=a{M}_2/M$ and $a_2=a{M}_1/M$. Summing their orbital angular momenta givesUsing Kepler's third law, ${\Omega }^2=GM/a^3$, also gives $J=M_1{M}_2\sqrt{Ga/M}$. Under conservative binary mass transfer, $M$ and $J$ stay fixed, so $a\propto (M_1{M}_2{)}^{-2}$. Since $d\log M_2/d\log M_1=-q$, the Roche-lobe radius response exponent isFor the supplied giant response, luminosity stays fixed during the mass-loss perturbation and ${\zeta }_{\ast }=d\log R/d\log M_1=-0.27$. The change of the logarithmic overfill is $({\zeta }_{\ast }-{\zeta }_L)d\log M_1$. Because the donor loses mass, this reduces overfill and stabilizes transfer precisely when ${\zeta }_{\ast }>{\zeta }_L$. Thus the stated model requiresThis is a response criterion within the given radius law; a different adiabatic donor response would change the dynamical threshold.

The initially more massive star evolves first. If it waits until it has the supplied giant structure while still having $q>1$, conservative overflow is unstable by this criterion. If Roche-lobe overflow instead begins while the donor is on the main sequence or shortly after central hydrogen exhaustion, its largely radiative-envelope response can allow transfer before a deep convective giant envelope develops. Mass loss and accretion can then reverse the mass ratio before it reaches the giant state. This is the binary mass-ratio reversal route to an Algol binary, through Case A mass transfer or early case B mass transfer.

At lobe filling, the lobe approximation and Kepler's law give the pre-giant Roche-lobe-filling period limitIt increases with the donor's radius. Requiring $R_L<R_{\mathrm{BGB}}=1.68R_{\odot }(M_1/M_{\odot }{)}^{5/3}$ therefore yieldsThe companion mass cancels in this lobe approximation.

Since $P\propto a^{3/2}$ at fixed total mass, the separation scaling gives $\boxed{P(M_1{M}_2{)}^3=\text{constant}}$. Use solar-unit masses $m_i=M_i/M_{\odot }$ and $m=m_1+m_2$, and let $K=P(m_1{m}_2{)}^3$. ThenThe derivative vanishes at $m_1=5m/8$, and its derivative is $5/m_1^2+3/(m-m_1{)}^2>0$, while the ratio diverges at both endpoints. Hence this is the global minimum, the conservative pre-giant period-ratio minimum.

For the stated Algol parameters, $m=4$ and $K=3(1\cdot 3{)}^3=81$ days. At the minimum $m_1=2.5,m_2=1.5$, the period is $81/(2.5\cdot 1.5{)}^3=1.536$ days and $P/m_1^2=0.24576$ days, well below $P_0=0.8$ days. This is a possible initially more-massive donor configuration with overflow before the giant branch. The current $q=1/3$ is also below the giant stability threshold. Thus Algol is compatible with the proposed conservative early-overflow route, although this necessary geometric test alone does not prove its full evolutionary history.

For RT Lac, $m=2.4$ and $K=5.1(0.9\cdot 1.5{)}^3$. Its minimum occurs at $m_1=1.5,m_2=0.9$, where the mass product equals its current value, so $P=5.1$ days andNo conservative antecedent on this orbit can begin overflow before the giant branch. Beginning with an initially more massive giant instead encounters the instability just derived. RT Lac cannot follow the stated coeval, conservative Algol-formation model. A plausible alternative is rapid initial transfer with substantial mass and angular-momentum loss, via an outflow, strong wind or envelope-ejection episode. Such nonconservative binary mass transfer changes both total mass and $J$ and removes the period-product constraint. Its present mass ratio $0.9/1.5=0.6$ permits stable current giant transfer. The given data do not determine a unique mass-loss history or establish a specific ejection mechanism.

Use a circular orbit, as in the stated Roche lobe approximation, with orbital angular speed $\Omega$ and negligible stellar spin. The center of mass distances are $a_1=a{M}_2/M$ and $a_2=a{M}_1/M$. Summing the two orbital angular momenta givesKepler's third law, ${\Omega }^2{a}^3=GM$, makes this $J=M_1{M}_2\sqrt{Ga/M}$. In conservative mass transfer, both $M$ and $J$ are fixed, while $d{M}_2=-d{M}_1$. ThusThe Roche-lobe radius response exponent is consequentlyDuring a dynamical mass-loss perturbation, the deep interior and luminosity do not have time to change. The supplied giant structure therefore has stellar radius response exponent ${\zeta }_{\ast }=-0.27$. Its fractional overfill changes by $d\ln (R/R_L)=({\zeta }_{\ast }-{\zeta }_L)d\ln M_1$. The binary mass ratio uses donor mass divided by accretor mass. Since $d\ln M_1<0$, self-limiting Roche-lobe overflow requires ${\zeta }_{\ast }>{\zeta }_L$. Hence the dynamical stability condition isAt equality the linear restoring response vanishes. This conservative mass-transfer critical mass ratio uses the given radius exponent, rather than imposing the different fully convective $-1/3$ approximation.

The initially more massive component normally evolves first. If it first overflowed only after acquiring the given giant response, it would have $q>1$ and the mass loss would be dynamically unstable: its radius grows while its Roche lobe initially contracts. Starting overflow earlier can avoid this outcome. A main sequence or early post-main-sequence donor star can have a radiative envelope with a stabilizing contraction response. Transfer then reverses the binary mass ratio before the donor develops the giant structure. This is the route to an Algol binary through Case A mass transfer during core hydrogen burning, or early case B mass transfer after core hydrogen exhaustion but before a deep giant envelope develops.

For the initial Roche-lobe overflow to occur before the base of the giant branch, its lobe must be smaller than the base-of-branch stellar radius. Combining $R_L=0.462a(M_1/M{)}^{1/3}$ with Kepler's third law cancels the companion dependence:Writing $m_1=M_1/M_{\odot }$, substitute the mass-radius relation to obtain the pre-giant period limit:We use the specified $0.8\mathrm{days}$ below. This is a pre-giant Roche-lobe-filling period limit, not a condition on the current stripped donor's radius.

At fixed total mass, Kepler's third law gives $P\propto a^{3/2}$, while fixed orbital angular momentum gives $a\propto (M_1{M}_2{)}^{-2}$. Therefore the conservative period invariant isTo find the smallest possible pre-giant period ratio, use solar-unit masses $m_1,m_2$ and fixed $m=m_1+m_2$. Let $K=P(m_1{m}_2{)}^3$. Along this period-product invariant for conservative mass transfer,The denominator has logarithmic derivative $5/m_1-3/(m-m_1)$, with a strictly negative derivative of its own. Its unique maximum, hence the unique minimum of $F$, occurs atSince $F$ diverges at either endpoint, this is the global minimum period-to-donor-mass ratio for conservative evolution.

For the quoted Algol binary, $K=3(1\times 3{)}^3=81\mathrm{days}$ and $m=4$. Choosing the minimizing progenitor gives $m_{1,i}=2.5$, $m_{2,i}=1.5$ andIts pre-giant upper limit is $0.8(2.5{)}^2=5\mathrm{days}$, comfortably above this initial period. Subsequent conservative mass transfer of $1.5M_{\odot }$ produces the stated masses and period. Thus the data permit the proposed early-overflow formation route.

For RT Lac, instead $K=5(0.5\times 1.5{)}^3=2.109375\mathrm{days}$ and $m=2$. At its most favorable conservative progenitor, $m_{1,i}=1.25$, $m_{2,i}=0.75$ andNo conservative progenitor meets the pre-giant overflow condition. An initially more massive giant donor also fails the dynamical-stability condition derived above. Within this stable Algol-type formation model, the current RT Lac system therefore requires nonconservative mass transfer.

A plausible history is substantial envelope loss from the system, through a stellar wind or escaping overflow, possibly with a common envelope episode. Escaping matter also carries orbital angular momentum, so the total-mass and period-product invariants no longer constrain its initial orbit. A more massive progenitor can then shed its envelope and reach the present low donor mass without requiring the excluded conservative sequence. The supplied final data do not determine a unique mass-loss or angular-momentum-loss history.