OWAO 2022, Теория — Задача 6. Orbital Time Machine

• $2 \times 10^{6}$ s

  τ = 10/(FA) = 2.0·10⁶ s ≈ 23 d.

First, let us estimate the distance to the star. For simplicity, we will assume here and below that $z \ll 1$ still:

We consider the star to be a black body. Average energy of a black-body photon is $\bar{\varepsilon} = 2.70k_BT$. The estimated photon flux density (number of photons per square meter per second) from the star in the Solar system is

Not all of these photons fall within the telescope’s wavelength coverage. The observed $\lambda$ range of 0.6 – 28.5 µm corresponds to initial $\lambda' = \frac{\lambda}{1+z}$ range of 0.5 – 23.75 µm.

Recalling Planck’s law and expression $\varepsilon = h\nu = \dfrac{hc}{\lambda}$ for energy of a photon, we introduce substitution $x \equiv \frac{hc}{\lambda k_BT_\odot}$:

Such a range covers both the mean value over the energy distribution ($x_{\mathrm{mean}} = 2.70$, mentioned before) and the rather sharp peak of the distribution itself ($x_{\max} = 2.82$). So, we can expect that in fact almost all photons fall within the JWST wavelength coverage (in fact, 90 % of the photons from the star meet that condition).

By the way, the frequency for maximal spectral radiance of a black body with temperature $T_\odot$ is $\nu_{\max} = 341$ THz with corresponding wavelength $\lambda'_{\max} = 0.88$ µm. The redshift only shifts this peak further into the infrared.

Note. Constants $k_B$ and $\sigma$ are not given in the table. They can be estimated using well-known energetic properties of the Sun. For example, the solar luminosity

or the solar constant

So, the Stefan–Boltzmann constant

One could also recall Wien’s displacement law:

where $\lambda_{\max} = 0.5$ µm for the solar spectrum. Of course, $\lambda_{\max} \neq \lambda'_{\max}$: it is well-known that parameterization by frequency implies a different maximal wavelength than parameterization by wavelength.

Then $x_i = \dfrac{\lambda_{\max}}{\lambda_i}\cdot 5.0$, and

Actually, the exact expression for $F$ in this model is

The last integral can be calculated numerically, although this was not required for the presented solution. One could also estimate that integral as

How long does it take for the telescope ($A = 25\ \mathrm{m^2}$) to accumulate 10 photons from the star?

Marking Scheme:

• Distance to the galaxy — 2 pt.
• Illuminance — 3 pt.
• Photon flux (taking into account the wavelength coverage and redshift) — 3 pt.
• Time interval — 2 pt.