OWAO 2024, Теория — Задача 2. Satellite Tracking

• a) (19° N, 39° E) or (19° S, 39° E)
• b) 47 °

a) Since the inclination is positive and does not exceed $90^{\circ}$, the orbital motion of the satellite occurs in the same direction as the rotation of the Earth around its axis. We should consider two cases, since it is not known whether the satellite rises above the plane of the equator or falls below the equator.

Let us consider the case when the satellite rises above the equator. First, we will determine by Kepler’s third law how many degrees of its orbit ($r = 3.0 \cdot 10^4$ km) the satellite will fly during the time $t = 3$ h:

The satellite has completed less than a quarter of its orbit.

Now we assume that the Earth does not rotate — we will take this effect into account when obtaining the final longitude of the second point.

Let $\tilde P_1$ and $\tilde P_2$ be the projections of positions of the satellites $S_1$ and $S_2$ on the non-rotating Earth (see Figure 2). The goal is to find the coordinates $(\varphi_2;\ \tilde\lambda_2)$ of the point $\tilde P_2$.

We draw a perpendicular from the point $\tilde P_2$ to the equator. Using spherical laws for $\triangle\tilde P_1\tilde P_2\tilde P_\perp$ one can write

The longitude $\tilde\lambda_2$ is determined unambiguously

In 3 hours, the Earth will turn in the same direction as the satellite, at an angle

Then the actual longitude of the point $P_2$ is $\lambda_2 = \tilde\lambda_2 - \Delta\lambda = 84^{\circ} - 45^{\circ} = 39^{\circ}\,\mathrm{E}$.

The latitude of $P_2$ is $\sin\varphi_2 = \arcsin(\sin\tilde P_1\tilde P_2 \cdot \sin i) = \arcsin(\sin 75^{\circ}\sin 20^{\circ}) = 19.3^{\circ}$.

Recalling that there are two cases of satellite motion (the starting point could be an ascending or descending node of the orbit), we write down the possible coordinates of point $P_2$:

b) First, we determine the angular distance between the observation points:

Next, we determine the distance from point $P_1$ to the satellite (point $S_2$) by the law of cosines (see Figure 3):

Finally, we determine the angle $OP_1S$:

We conclude that from the point $P_1$, the satellite is observed at the altitude of $137^{\circ} - 90^{\circ} = 47^{\circ}$.

Marking Scheme:

• Question (a)

  1. Arc $S_1S_2$ that the satellite passes in 3 hours — 2 pt.
  2. Geometric model, mentioning two cases — 2 pt. + 1 pt.
  3. Expressions for $(\varphi_2;\ \tilde\lambda_2)$ — 4 pt.
  4. Taking into account the rotation of the Earth — 3 pt.
  5. Coordinates of $P_2$ — 2 pt.
• Question (b) 6. Geometric model — 2 pt. 7. Angular distance between $P_1$ and $P_2$ — 1 pt. 8. Distance $P_1S_2$ — 1 pt. 9. Altitude of $S_2$ at $P_1$ — 2 pt.