Transit simulation

I simulated the transit of your spaceship in front of a Sun-like star. Besides the dimensions of your ship, I made the following assumptions:

• A roughly circular orbit, with semi-major axis $a=0.1\text{ AU}$.
• A limb darkening coefficient of $u=0.6$, and a linear limb-darkening law.
• The star is Sun-like in terms of mass, luminosity, and radius.

Mathematically, I used a limb darkening law of the form $$I(r)=I_0(1-u(1-\sqrt{\frac{R_{\ast }^2-r^2}{R_{\ast }^2}}))$$ where $I$ is the power per unit area on the star. I calculated the proportionality constant $I_0$ such that the star's luminosity before the transit was equal to $L_{\odot }$. For each time $t$ during the transit (with 500 timesteps), I created a 50-by-100 grid representing the outline of the ship, and integrated $I(r)$ across that grid, then subtracted this from $I_0$.

Here's the resulting light curve I simulated:

A quantity of interest is $\mathrm{\Delta }F/F$, the fractional change in flux at a point during the transit. I found the maximum value of $\mathrm{\Delta }F/F$ to be $4.14\times {10}^{-9}$ - higher than Joe Kissling's answer of $1.3\times {10}^{-10}$. The reason for this, I think, is that Joe used the average intensity of the Sun; taking limb darkening into account means that the center of the solar disk is brighter than average, meaning more light is blocked at the peak of the transit, and less light is blocked at the beginning and end.

Lost in the noise

There are two factors that determine whether or not a given value of $\mathrm{\Delta }F/F$ can be measured by a telescope. The first consideration is that a star's luminosity can vary on timescales of hours or days. Even a non-variable star like the Sun may experience changes of $\sim {10}^{-5}$ - orders of magnitude larger than our value. Sunspots, for instance, can be contributors. Here, stellar variability is going to wash out our transit.

The second issue is that a telescope's sensitivity is limited. Hubble and Kepler can view $\mathrm{\Delta }F/F\sim {10}^{-4}$ - impressive, but not good enough for our purposes. For comparison, a hot Jupiter orbiting a Sun-like star may have $\mathrm{\Delta }F/F\sim {10}^{-2}$, and an Earth-like planet orbiting a Sun-like star may have $\mathrm{\Delta }F/F\sim {10}^{-6}$.

In short, even at the peak of the transit, the change in flux is going to be several orders of magnitude too low for current (or even near future) telescopes to detect, and it will likely be lost in regular fluctuations in the star's luminosity.

posted over 6 years ago

HDE 226868‭

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