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Section 9: Astronomy and Astrophysics

1. Rotational Velocity

Calculate the linear speed (in km/s) of a point on the Earth's equator due to its rotation. Earth's radius \(\approx 6378\) km.

2. Orbital Mechanics

Calculate the orbital speed of the International Space Station (ISS), which orbits at an altitude of approximately 400 km above the Earth's surface. (Earth's mass \(M_E \approx 5.97 \times 10^{24}\) kg). Compare this speed of Earth's orbital speed around the Sun (Earth-Sun distance \(\approx 150 \times 10^6\) km, Earth's orbital period \(\approx 365.25\) days). Which is faster, the ISS around the Earth or Earth in its orbit around the Sun?

3. Microgravity

What is the acceleration due to gravity (\(g\)) at the altitude of the ISS (400 km)? Why do astronauts experience a state of "weightlessness" despite this gravity?

4. Geostationary Orbit

Satellites in geostationary orbit remain above the same point on Earth. What must their orbital period be? Calculate the altitude of a geostationary orbit above the Earth's surface.

5. Escape Velocity

What is the escape velocity from the surface of the Moon? (Moon's mass \(M_M \approx 7.35 \times 10^{22}\) kg; whereas Moon's radius \(R_M \approx 1,737\) km). Express the result in km/s and as a fraction of Earth's escape velocity (Earth's escape velocity \(\approx 11.2\) km/s).

6. Solar Gravity

Calculate the acceleration due to gravity on the surface of the Sun. By what factor would your weight increase if you could stand on its surface? (Sun's mass \(M_S \approx 2 \times 10^{30}\) kg; Sun's radius \(R_S \approx 6.96 \times 10^8\) m).

7. Megastructures

A "Dyson Sphere" is a hypothetical megastructure that completely encompasses a star to capture its energy output. If the mass of Mercury (\(3.3 \times 10^{23}\) kg) were used to build a solar panel sphere with a surface density of 10 kg/m², what would be the radius of the sphere?

8. Interplanetary Travel

How much time would it take to get from Earth to Mars when Mars is at its closest approach (55 million km)

a) for a message sent at the speed of light?

b) by a spacecraft traveling at a constant speed of 40,000 km/h = 11.11 km/s?

c) by "airplane" traveling at a constant speed of 900 km/h, typical for intercontinental flights?

9. Size and distance of the Sun

Aristarchus from Samos (310–230 BC) was an ancient Greek astronomer who attempted to determine the relative distances and sizes of the Sun and the Moon using geometric methods based on observations of lunar phases.

At the exact half-Moon phase (dichotomy), the Earth–Moon–Sun system forms a right triangle with the right angle at the Moon. The angular separation between the Sun and the Moon observed from Earth at that moment is \(\theta = 89.85^\circ\). The apparent angular diameter of both the Sun and the Moon is \(\alpha = 0.53^\circ\). The average Earth–Moon distance is \(d_{EM} = 3.84 \times 10^5\,\text{km}\).

Calculate:

  1. The Earth–Sun distance \(d_{ES}\) in km.

  2. The true diameter of the Sun \(D_S\) in km (use the small-angle approximation \(\alpha \approx D/d\), with \(\alpha\) in radians).

  3. The ratio of true diameters \(\frac{D_M}{D_S}\).

  4. How much the value of \(d_{ES}\) changes if \(\theta = 89.75^\circ\) (a historically quoted value attributed to Aristarchus from Samos) is used instead of \(89.85^\circ\). Briefly comment on the sensitivity of the result to the angle measurement and what this implies for Aristarchus’ method.

10. Measuring the Height of the Atmosphere

A medieval astronomer in Al-Andalus, Al-Zarqali (Arzachel) (1029–1087 AD), attempted to estimate the height of the Earth’s atmosphere using a geometric method based on sunset timing. He measured the interval between sunset and the moment when faint stars first became visible, assuming that this corresponds to the Sun reaching a true geometric depression angle \(\phi\) below the horizon. A chronicle reports that on a clear evening the time between sunset and the first appearance of faint stars was \(t = 40\) minutes.

Assume: - Earth radius \(R_E = 6370\,\text{km}\), - Earth’s rotation rate: full rotation in 24 hours (i.e., \(360^\circ\) in 24 hours), - A simple “sharp-edge atmosphere” model in which a Sun ray reaches the observer by just grazing the top of the atmosphere, giving

\[\cos\phi = \frac{R_E}{R_E+h}.\]

Find:

  1. The solar depression angle \(\phi\) (in degrees) implied by the measured time \(t\).

  2. The atmospheric height \(h\) in km.