OWAO 2025, Теория — Задача 3. Swing of the Spheres

• a) $r\approx4.9\ \mathrm{pc}$.
• b) $D\sim3\cdot10^2\ \mathrm m$.
• c) $m\approx3.3^{\mathrm m}$; no matching G2V star at this distance.
• d) $\mu=-10.6^{\mathrm m}/\mathrm{arcsec^2}$.
• e) Northern: $+43.8^\circ$ to $+90^\circ$; southern: $-43.4^\circ$ to $3.4^\circ$.

a) The maximum aberrational displacement is determined by the Earth’s full orbital velocity and is given by $\theta=v_\oplus/c=10^{-4}\ \mathrm{rad}=20.6''$. Although one could use the Earth’s velocity at perihelion to account for orbital ellipticity, this refinement is not essential for this calculation.

In this case, the parallactic displacement is $0.206''$, which corresponds to a distance of

b) To resolve details on the star’s disk, the telescope’s angular resolution must be finer than the star’s angular diameter. For a rough estimate, we can require the resolution to be better than approximately 0.2 times the angular size. Participants may use any reasonable factor for this estimation.

The angular resolution of an optical interferometer is given by:

where $\lambda$ is the wavelength of observed radiation and $D$ is the baseline (the distance between the telescopes).

The angular diameter $d$ of the star is related to its physical radius $R$ and distance $r$:

Equating the required resolution to the interferometer’s resolution gives:

Using an observation wavelength of 550 nm (visible spectrum characteristic wavelength — green light), we can solve for the required baseline $D$:

This calculated distance is similar to the maximum baseline of the CHARA optical interferometer.

c) The distance to the object is relatively small, so interstellar extinction is negligible. The relation between the apparent magnitude $m$, absolute magnitude $M$, and distance $r$ (in parsecs) is

The object described would be a bright, Sun-like star. However, no stars fitting this description are observed at the calculated distance of 4.9 pc. Most nearby stars are cooler than the Sun, and the well-known Alpha Centauri system is significantly closer. The star with parameters most similar to those in this problem is $\tau$ Ceti: it is somewhat cooler (spectral type G8V) and slightly closer (3.7 pc) than the 4.9 pc distance we obtained.

d) To estimate the surface brightness of the star’s disk, we neglect limb darkening and assume uniform brightness across its surface. Under this assumption, the apparent magnitude of the entire disk $m$ can be related to the apparent magnitude per square arcsecond $\mu$ using Pogson’s formula:

The angular diameter of the disk, expressed in terms of its physical diameter and distance, is converted to arcseconds as follows:

Alternative way. It may be recalled that, in the absence of absorption, the surface brightness of objects does not depend on distance. The desired quantity is equal to the surface brightness of the solar disk. The apparent solar diameter is approximately $31'=1860''$:

e) The eccentricity of the aberration ellipse is related to the ecliptic latitude of the star. The minor axis of the aberration ellipse is related to the semi-major axis $a=\theta$ as $b=\theta\sin\beta$, whence the eccentricity is

Consequently, for an eccentricity of $e=0.4$, the ecliptic latitude is $\beta=\pm\arccos0.4\approx\pm66.4^\circ$.

The ecliptic longitude of the star is unspecified. Consequently, we have to estimate the possible range of altitudes at upper culmination using the known ecliptic latitude $\beta$ and geographic latitude $\varphi=43.6^\circ$. To determine the corresponding declination $\delta$, which is required for the altitude calculation, we apply the spherical law of cosines:

The minimum declination occurs when the star is located near the winter solstice point ($\sin\lambda=-1$), and the maximum declination occurs near the summer solstice point ($\sin\lambda=+1$).

We now estimate the culmination altitudes. For the northern celestial hemisphere, the observer’s latitude falls within the calculated range of declinations, meaning the star’s declination can equal the latitude. This implies the star can pass directly through the zenith (altitude $h=90^\circ$) at upper culmination. To find the minimum possible altitude at upper culmination, we use the extreme declination values:

Thus, the altitude at upper culmination ranges from $+43.8^\circ$ to $+90^\circ$.

For the southern celestial hemisphere,

Thus, the altitude at upper culmination ranges from $-43.4^\circ$ to $3.4^\circ$.

Marking Scheme:

• Question (a) Distance

  1. Estimate of maximum aberration — 1 pt.
  2. Parallax — 1 pt.
  3. Distance from parallax — 2 pt.

  If the distance is incorrect, subsequent items are not penalized if it is used further.
• Question (b) Interferometer baseline

  1. Angular size of the star — 2 pt.
  2. Baseline length — 2 pt.
• Question (c) Apparent magnitude

  1. Comparison with the Sun, using the apparent magnitude — 1 pt.
  2. Apparent and absolute magnitude — 2 pt.
  3. No Sun-like stars exist at such a distance — 2 pt.
• Question (d) Surface brightness — 2 pt.
• Question (e) Altitudes of upper culmination

  1. Ecliptic latitude from ellipse eccentricity (recalling or deriving) — 2 pt.
  2. Range of possible declinations — 1 pt.
  3. Altitudes of upper culmination — 2 pt.

Source note: The official key on page 9 prints a minus sign in the first solved expression for $\mu$, although the preceding Pogson relation and its alternative solar-disk calculation imply a plus sign. Both printed equations and the printed result are retained. On page 10, the southern-case $\delta_{\min}$ and $\delta_{\max}$ labels in the altitude equations use the opposite endpoint values from the immediately preceding table; the original labels, values and resulting range are retained.