Would the impact of a planet on its host star yield a noticeable brightening if it happened to occur on the side of the star facing you?

First, can a planet impact its host star? Yes, if it is solid and more than twice as dense as the star. So only the cores, not the gaseous envelopes, of hot Jupiters are likely to impact their host stars (the envelopes will be tidally stripped first). If the planet is molten then some or all if it might be tidally stripped before impact. But as long as the stripped material stays high density it should produce the same luminosity from shocked gas on the star's surface, just more spread out over time, as the tidally stripped material will be spread out before and behind the planet in its orbit.

A way to estimate the impact luminosity is to take a spherical planet of a certain density and radius R_p, give it in a Keplerian azimuthal velocity at the photosphere of a star of radius R_s and temperature T_eff, and some radial velocity. Treating the stellar photosphere as a uniform-density slab of gas, work out the post-shock temperature of the obliquely shocked gas and the timescale t for the planet to move its own diameter into the star as its azimuthal velocity slows. (I am assuming that the tau=1 distance in the stellar atmosphere is << R_p). The radiating area of the shocked gas is roughly A=2*R_p*v_phi*t (for v_phi*t < R_s), and the fractional luminosity increase from the shock-heated gas is dL=[A/(pi*R_s^2)]*[(T_shock^4/T_eff^4)-1]. Taking a maximal case of A=2*R_p*R_s, dL~(R_p/R_s)*(T_shock^4/T_eff^4) where I've assumed T_shock >> T_eff. For Earth, R_p/R_s=0.01, so the effect can be large in principle. Even if you scale back and assume A ~ R_p^2, the dependence on the fourth power of the temperature means it might be a big flash.

Next step: refine the analysis. Suppose that the planet has sunk beyond sight by the time it has lost a fraction X of its original momentum (assuming v_phi >> v_rad, then it has slowed from v_phi to (1-X)v_phi). After that time t an energy (M_p/2)*(Xv_phi)^2 has been converted into heat. Thermal energy E_th deposited by planet is 3/2 n V k_B (T_shock-T_eff) where n is pre-shock density at stellar surface and V is volume swept out by planet before it is too deep in the Sun to be seen (I am taking this to be the point at which the planet's center is at R=R_s-R_p): V = 1/2 \pi R_p^2 v_phi(1-X/2) t, so E_th = 3(1-X/2)/4 n \pi R_p^2 v_phi t k_B (T_shock-T_eff).

Equating the lost kinetic energy E_k to the gained thermal energy,

T_shock-T_eff = M_p X^2 v_phi^2 * 2 / (1-X/2)*2 n \pi R_p^2 v_phi t k_B, or roughly

T_shock-T_eff = M_p X^2 v_phi / \pi k_B n R_p^2 t, which is ~4x low for X~1.

For a Keplerian orbit at the stellar surface, v_phi=sqrt(GM_s/R_s). In that case:

T_shock-T_eff \propto M_p M_s^1/2 X^2 / R_s^1/2 R_p^2 n t.

The timescale for shocked gas to radiatively cool from T_shock to T_eff is the ratio of the deposited thermal energy to the radiative flux (~ A \sigma_SB T_shock^4):

t_cool \propto (\pi/2) n R_p^2 v_phi t k_B dT / 2 R_p v_phi t \sigma_SB T_shock^4

t_cool \propto (\pi/4) n R_p k_B dT / \sigma_SB T_shock^4 \propto R_p/T_shock^3 for T_shock >> T_eff. Ignoring conduction overestimates the length of the luminous phase. The conduction timescale depends on T_shock-T_eff, so conduction will hasten the initial dropoff in luminosity but yield a longer tail at slightly elevated luminosity.

So with E_k \propto M_p X^2 M_s / R_s (assume a Keplerian v_phi) and

t_cool \propto R_p^7 R_s^3/2 n^3 t^3 / M_p^3 M_s^3/2 X^6, the peak luminosity will be roughly E_k/t_cool, or L_peak \propto M_p^4 v_phi^5/2 X^8 / R_p^7 n^3 t^3. The more massive the planet, the brighter the flash. The faster the planet is orbiting on impact, the brighter the flash. The more momentum is dumped before the planet sinks beyond sight, the brighter the flash. The smaller the planet (at fixed mass), the brighter the flash (the same momentum is dumped into a smaller volume of the star). The shorter the timescale over which momentum/energy is transferred, the brighter the flash. (For "brighter" above read "more luminous".)

The only remaining free parameters are t and X. We need two equations involving them to eliminate them; the first is momentum conservation. The planet's initial momentum equals the total momentum of planet plus stellar gas at the point where the planet disappears from sight (the stellar gas is pushed aside, but immediately after impact has velocity ~v_phi): M_p v_phi = (M_p + mu_s n V) X v_phi

where mu_s is the mean mass per particle in the star's atmosphere and the star's rotational velocity is ignored. This becomes a quadratic equation for X which involves t: M_p = (M_p + mu_s n \pi R_p^2 \sqrt{GM_s/R_s} (1-X/2) t / 2) X.

The second equation comes from approximating the planet's trajectory as an elliptical orbit with apoapsis at R_s and velocity v_ap=(1-X/2)sqrt(GM_s/R_s) at that apoapsis. Those two parameters determine the eccentricity e:

v_ap=\sqrt{(1-e)GM_s/R_s} so (1-X/2)^2 = (1-e). Approximate t as twice the time for an object in such an elliptical orbit to move from R=R_s to R=R_s-R_p.

Given an initial guess for X, that time can be computed from equal areas in equal times + equations for the arc length and circumference of an ellipse of known e and apoapsis.

Once t is found, it can be plugged into the quadratic to yield a new value for X. The process can be iterated until (hopefully) it converges.

Knowing t and X should give you the absolute values and scalings of t_cool and L_peak, dependent only on R_p, M_p, R_s, M_s, n, mu_s, T_eff. One can also find the peak magnitude increase from -2.5*log10[(A/\piR_s^2)(T_shock/T_eff)^4]. However, some of those dependences will be non-analytic, subsumed in the estimation of t. (Complicated analytic approximations for the arc length and circumference of ellipses do exist, so complicated analytic approximations for t_cool and L_peak could be given. But my estimation of L_peak=E_th/t_cool is rather crude, so before giving complicated expressions I would improve the existing analysis. E.g. by allowing for non-negligible radial velocity [elliptical orbit from the start].)

L_peak will also have an inclination angle correction: the shocked gas will emit isotropically, but its projected area will depend on its location on the side of the stellar surface facing you. If the planet/stellar atmosphere density contrast is large enough that the length of the strip of the star's surface affected by the planet is >~ R_p, the inclination correction will get complicated.

[Possibly to be continued at some point when/if I feel like working out the numbers for an Earth analogue impacting a G2V star.]

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