= Solution
Write $R_p$ for the physical planet radius and $R_c=\eta R_{H1}$ for the debris-cloud radius; the source uses $R_1$ for both even though their stated size orderings are opposite. A central <transit duration of a small planet> on a nearly circular <Kepler orbit> is $D_p\simeq2R_\star/(n_1a_1)=P_1R_\star/(\pi a_1)$. <Kepler's third law>, with $M_\star=4\pi\rho_\star R_\star^3/3$, gives
$$
\frac{P_1}{D_p^3}\simeq\frac{\pi^2G\rho_\star}{3}.
$$
A noncentral planetary <exoplanet transit> is shorter, so within the small-radius and nearly circular approximations, a duration requiring
$$
\boxed{\frac{P_1}{D_1^3}<\frac{\pi^2G\rho_\star}{3}}
$$
is too long for the planet alone and favors extended debris. For a central cloud transit with $R_c\gg R_\star$, the path length is approximately $2R_c$. Using the <Hill radius> $R_{H1}=a_1(M_1/(3M_\star))^{1/3}$ gives $D_1/P_1\simeq R_c/(\pi a_1)$, hence the <Hill-cloud transit mass estimate>
$$
\boxed{\frac{M_1}{M_\star}\simeq3\left(\frac{\pi D_1}{\eta P_1}\right)^3.}
$$
This equality assumes a central cloud chord. An orbit described merely as nearly edge-on need not have central cloud crossings. If its projected cloud impact distance is $s$, the measured half-chord is $\sqrt{R_c^2-s^2}$, so the boxed mass is multiplied by $[1-(s/R_c)^2]^{-3/2}$. It is a lower estimate when that geometry is unknown.
For the dynamical calculation use $\mu_2=M_2/M_\star$, $A=GM_2/a_2$, and $\mathcal R=A f(\alpha)e_1\cos\phi_1$. The derivatives in the supplied <Lagrange planetary equations> hold the other <osculating orbital elements> fixed:
$$
\partial_{\lambda_1}\mathcal R=jAf e_1\sin\phi_1,
\quad \partial_{a_1}\mathcal R=\frac A{a_2}f'e_1\cos\phi_1,
\quad \partial_{e_1}\mathcal R=Af\cos\phi_1.
$$
Since $A/(n_1a_1^2)=n_1\mu_2\alpha$, the physical <mean-longitude equation for a first-order resonant term> is
$$
\boxed{\dot\lambda_1=n_1[1+\mu_2g(\alpha)e_1\cos\phi_1],\qquad
g(\alpha)=\frac\alpha2f(\alpha)-2\alpha^2f'(\alpha).}
$$
The source's epoch notation needs care when $n_1$ varies. Literally differentiating $\lambda_1=n_1(t)t+\epsilon_1(t)$ adds $t\dot n_1$; one cannot retain that definition and also identify the printed $\dot\epsilon_1$ with $\dot\lambda_1-n_1$. A consistent version writes $\lambda_1=\int_0^t n_1(s)\,ds+\epsilon_{\rm int}$ and assigns the displayed correction to $\dot\epsilon_{\rm int}$. Equivalently, the canonical <mean-longitude equation for a first-order resonant term> follows from the <disturbing function> directly. The result for $g$ is the intended physical equation under that convention.
Away from <resonant-argument libration>, neglect changes in $e_1,\alpha,\varpi_1$ on one slow cycle to this order. With $n_0=2\pi/P_1$ the reference <mean motion>, the signed slow frequency is
$$
\omega=(j+1)n_2-jn_0=-\frac{jn_0\Delta}{1+\Delta},\qquad
P_t=\frac{2\pi}{|\omega|}=\frac{P_1(1+\Delta)}{j|\Delta|}.
$$
This is the <near-resonant transit-timing superperiod>. Choose $\phi_1(0)=0$. Differentiating $n_1\propto a_1^{-3/2}$ with the <semi-major axis> equation gives
$$
\dot n_1=-3j\mu_2n_0^2\alpha f e_1\sin(\omega t).
$$
Let $C=3j\mu_2n_0^2\alpha f e_1$ and $n_s=n_1(0)$. Direct <integration> gives
$$
n_1(t)=n_s+\frac C\omega[\cos(\omega t)-1],
\qquad
\lambda_1(t)=\left(n_s-\frac C\omega\right)t
+\left(\frac C{\omega^2}+\frac{n_0\mu_2g e_1}{\omega}\right)\sin(\omega t),
$$
with zero initial <mean longitude>. The constant and linear terms are absorbed into the fitted transit ephemeris: the measured mean frequency is $\bar n=n_s-C/\omega$. This distinction avoids mistaking an arbitrary initial frequency for the long-term fitted period.
For $|\Delta|\ll1$, the double-integrated frequency term dominates the direct $g$ term provided $g/f$ stays finite. Relative to the fitted linear ephemeris, $\lambda_1(t_k)=2\pi k$ gives
$$
t_k\simeq P_1k+A_t\sin(\omega t_k),\qquad
A_t=-\frac{3j\mu_2n_0\alpha f e_1}{\omega^2}
\simeq-\frac{3\mu_2\alpha f e_1}{jn_0\Delta^2}.
$$
Here $P_1$ denotes the fitted mean period at the retained order. The signed-frequency convention fixes the phase sign. Replacing $\omega$ by $2\pi/P_t>0$ changes the sign of the signed $A_t$ when needed; a nonnegative amplitude has
$$
\boxed{\frac{|A_t|}{P_t}\simeq\frac3{2\pi}\mu_2\frac{\alpha|f(\alpha)|e_1}{|\Delta|}.}
$$
Using a signed $P_t=2\pi/\omega$ instead gives the source's signed proportionality $A_t/P_t\simeq3\mu_2\alpha f e_1/(2\pi\Delta)$. The leading implicit transit equation can equally use $P_1k$ inside the sine, since the difference is higher order in the perturbation. This is the <eccentricity-enhanced near-resonant transit-timing variation>; its derivation assumes small longitude oscillations, $\mu_2e_1\ll\Delta^2$ at fixed resonance coefficients, and the effectively fixed eccentricity used in this truncated model. Dominance over forced-eccentricity timing terms typically also requires $e_1\gg|\Delta|$ at fixed $j$. A full transit calculation also includes forced <orbital eccentricity> and the conversion from <mean longitude> to the actual sky-crossing angle. Those are lower order in this eccentricity-enhanced regime, rather than identically absent; broader near-resonant formulas are derived by https://arxiv.org/abs/1207.4192[Lithwick, Xie and Wu].
Inside a <mean-motion resonance>, $\phi_1$ undergoes <resonant-argument libration> and its evolution must be solved together with $a_1,e_1,\varpi_1$. The constant circulation frequency $\omega$ and the associated $\Delta^{-2}$ expansion then fail. At approximately fixed eccentricity, the resonant <resonant pendulum approximation> has libration frequency of order $jn_0\sqrt{\mu_2\alpha|f|e_1}$, up to numerical factors. The <transit-timing variation> follows that libration period and depends on its <libration amplitude of a resonant argument>. Exactly at a resonant equilibrium there need be no libration signal; near a <separatrix> the period grows, and eccentricity dynamics can modify the simple estimate. Resonant amplitudes remain controlled by the bounded resonant motion rather than diverging as the circulating detuning tends to zero.
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