At what distance are the two farthest visible stars of the Big Dipper - with the astronomical designation $\alpha \mathrm{U}$ Ma and $\eta \mathrm{U}$ Ma - if their distances from the Sun are 59 and 200 light-years, respectively, and their coordinates, analogous to geographical latitude and longitude, the so-called declination and right ascension, are as follows:

$$
\begin{array}{lll}
\text{for } \alpha \mathrm{U} \text{ Ma:} & \delta=+62^{\circ} 9^{\prime}, & \alpha=10^{\mathrm{h}} 59^{\mathrm{m}} \\
\text{for } \eta \mathrm{U} \text{ Ma:} & \delta=+49^{\circ} 41^{\prime}, & \alpha=13^{\mathrm{h}} 45^{\mathrm{m}}
\end{array}
$$

Calculate the distance between the two farthest visible stars of the Little Dipper as well. The data:
$\alpha$ U Mi: distance: $\beta$ U Mi: distance: $\quad 96$ light-years, $\quad \delta=74^{\circ} 28^{\prime}, \quad \alpha=14^{\mathrm{h}} 51^{\mathrm{m}}$.[^0]

[^0]: ${ }^{1}$ The declination is the angular distance of a star from the celestial equator, which is the projection of the Earth's equator onto the celestial sphere from the center of the Earth; the declination is measured on the great circle of the celestial sphere that passes through the star and the Earth's axis of rotation, the so-called celestial axis. The right ascension is the angular distance between the two points where the mentioned great circle and the chosen great circle intersect the celestial equator; these great circles - similar to terrestrial meridians - are actually semicircles, so the mentioned intersection points are uniquely determined. The angular distance is not measured along the shorter arc of the equator, but from the intersection point of the chosen great circle, eastward from $0^{\circ}$ to $360^{\circ}$, or - since right ascension is typically given in time units (stellar time) - from $0^{h}$ to $24^{h}$.

    For the solution, we can use the solution of problem 1295 in K. M. L. 29 (1964) p. 138. We can neglect the proper motion of the Earth, the Sun, and the stars involved in this problem, and we can take the celestial axis as a straight line passing through the center of the Sun and parallel to the Earth's axis, and the plane of the celestial equator as the plane perpendicular to the celestial axis and passing through the center of the Sun.

    The given coordinates are from the following source: Wodetzky József: Orientation in the Starry Sky (TTIT, Budapest 1955), and the distances are from: Kulin-Kolbenheyer: The World of Stars (Budapest, 1947). The interpretation of the abbreviations: $\alpha$ U Ma: alpha ursa majoris = the star with the designation $\alpha$ in the constellation Ursa Major; U Mi: ursa minor = Little Dipper.
![](https://cdn.mathpix.com/cropped/2024_05_02_f499ad2007642742a8c1g-1.jpg?height=943&width=1225&top_left_y=117&top_left_x=445)

In the triangle defined by Earth (F) and the two stars ($C_{1}$ and $C_{2}$), we know the sides $F C_{1}$ and $F C_{2}$, and from the given coordinates, we can calculate the cosine of the angle between them using the formula

$$
\cos \vartheta = \sin \delta_{1} \sin \delta_{2} + \cos \delta_{1} \cos \delta_{2} \cos \left(\alpha_{2} - \alpha_{1}\right)
$$

The right ascension differences for the two star pairs are:

U Ma pair: in hours and minutes $2^{\mathrm{h}} 46^{\mathrm{m}}$, in degrees $41.5^{\circ}$,

U Mi pair: in hours and minutes $13^{\mathrm{h}} 17^{\mathrm{m}}$, in degrees $199.25^{\circ}$,

thus, calculating the terms of (1) using logarithms for the first pair:

$$
\begin{aligned}
& \lg \sin 62^{\circ} 9^{\prime} = 9.9465 - 10 \quad \lg \cos 62^{\circ} 9^{\prime} = 9.6694 - 10 \\
& \lg \sin 49^{\circ} 41^{\prime} = \frac{9.8822 - 10}{0.8287 - 1} \quad \lg \cos 49^{\circ} 41^{\prime} = 9.8110 - 10 \\
& \lg \cos 41.5^{\circ} = \frac{9.8745 - 10}{0.3549 - 1}
\end{aligned}
$$

$\cos \vartheta_{1} = 0.6740 + 0.2264 = 0.9004$

For the second pair:

$$
\begin{array}{lrl}
\lg \sin 88^{\circ} 54^{\prime} = 9.9999 - 10 & \lg \cos 88^{\circ} 54^{\prime} = 8.2832 - 10 \\
\lg \sin 74^{\circ} 28^{\prime} = \frac{9.9838 - 10}{0.9837 - 1} & \lg \cos 74^{\circ} 28^{\prime} = 9.4278 - 10 \\
\lg \cos 160.75^{\circ} \| = \underline{9.9750 - 10} \\
& & 0.6860 - 3
\end{array}
$$

$\cos \vartheta_{11} = 0.9632 - 0.0049 = 0.9583$

Thus, for the two distances, applying the cosine rule:

$$
\begin{aligned}
& d_{1}^{2} = 59^{2} + 200^{2} - 2 \cdot 59 \cdot 200 \cdot 0.9004 = 2.22 \cdot 10^{4} \\
& d_{2}^{2} = 325^{2} + 96^{2} - 2 \cdot 325 \cdot 96 \cdot 0.9583 = 5.50 \cdot 10^{4}
\end{aligned}
$$

Therefore, the distance between the two farthest visible stars in the Göncöl Szekér case is $d_{1} \approx 149$ light-years, and in the Kis Göncöl case, it is $d_{2} \approx 235$ light-years.

Zsuzsa Cserháti (Székesfehérvár, Teleki B. g. III. o. t.)

Note. We used formula (1) in succession by referring to problems 1295, 1214, 1146, and 1045. In this regard, we present a proof different from the one found in the earliest reference.[^0]

Let the geographical longitude of $P_{i}$ be $\lambda_{i}$, and the latitude be $\varphi_{i}$, where $i=1,2$ and $-180^{\circ} \leq \lambda_{i} \leq 180^{\circ}$ (western, or eastern longitude) and $-90^{\circ} \leq \varphi_{i} \leq 90^{\circ}$ (southern, or northern latitude), and let us find the angle $\vartheta$ between the radii $O P_{1}$ and $O P_{2}$ of the Earth's sphere. Let the projection of $P_{i}$ on the Equator plane be $P_{i}^{\prime}$, and take the Earth's radius as the unit of length. Then

$$
O P_{i}^{\prime} = \cos \varphi_{i}, \quad P_{i} P_{i}^{\prime} = \sin \varphi_{i}
$$

(the latter is negative for points on the southern hemisphere). Applying the cosine rule to the triangle $O P_{1}^{\prime} P_{2}^{\prime}$,

$$
P_{1}^{\prime} P_{2}^{\prime 2} = \cos^{2} \varphi_{1} + \cos^{2} \varphi_{2} - 2 \cos \varphi_{1} \cos \varphi_{2} \cos \left(\lambda_{2} - \lambda_{1}\right)
$$

Thus, for the chord $P_{1} P_{2}$ of the spherical triangle $P_{1} P_{1}^{\prime} P_{2}^{\prime} P_{2}$, and then for the angle at $O$ in the triangle $O P_{1} P_{2}$,

$$
\begin{gathered}
P_{1} P_{2}^{2} = P_{1}^{\prime} P_{2}^{\prime 2} + \left(P_{1} P_{1}^{\prime} - P_{2} P_{2}^{\prime}\right)^{2} = \left(\cos^{2} \varphi_{1} + \sin^{2} \varphi_{1}\right) + \left(\cos^{2} \varphi_{2} + \sin^{2} \varphi_{2}\right) - \\
- 2 \cos \varphi_{1} \cos \varphi_{2} \cos \left(\lambda_{2} - \lambda_{1}\right) - 2 \sin \varphi_{1} \sin \varphi_{2} \\
\cos \vartheta = \frac{O P_{1}^{2} + O P_{2}^{2} - P_{1} P_{2}^{2}}{2 O P_{1} \cdot O P_{2}} = \sin \varphi_{1} \sin \varphi_{2} + \cos \varphi_{1} \cos \varphi_{2} \cos \left(\lambda_{2} - \lambda_{1}\right)
\end{gathered}
$$

László Cser (Csorna, Hunyadi J. g. III. o. t.)


[^0]:    ${ }^{2}$ K. M. L. $22(1961) 157$. o.