Astronomy olympiad practice problems with worked solutions for orbits, magnitudes, stars, and cosmology

If you are searching for astronomy olympiad practice problems, the fastest way to improve is to solve a small set of standard problems until the method feels familiar. The ten examples below cover orbits, magnitudes, stars, and cosmology, with each solution built around one key equation.

For a wider map of the syllabus, see the IOAA syllabus explained topic by topic, the astronomy olympiad preparation guide, the ten equations every astronomy student should know, and the unit on The Sky and Celestial Mechanics.

Orbits and celestial mechanics

1. A satellite in low Earth orbit

A satellite moves in a circular orbit 400 km above Earth. Find its orbital speed. Take Earth’s mass as \(5.97 \times 10^{24}\,\mathrm{kg}\) and Earth’s radius as \(6.37 \times 10^{6}\,\mathrm{m}\).

Key equation: \(v = \sqrt{GM/r}\)

Solution: The orbital radius is \(r = 6.37 \times 10^{6} + 4.00 \times 10^{5} = 6.77 \times 10^{6}\,\mathrm{m}\). Using \(G = 6.67 \times 10^{-11}\,\mathrm{SI}\),

\(v = \sqrt{(6.67 \times 10^{-11})(5.97 \times 10^{24})/(6.77 \times 10^{6})} \approx 7.67 \times 10^{3}\,\mathrm{m\,s^{-1}}\).

The answer is about \(7.7\,\mathrm{km\,s^{-1}}\).

2. The period of a planet

A planet orbits a Sun-like star at 4 AU. Find its orbital period in years.

Key equation: \(P^2 = a^3\) when \(P\) is in years and \(a\) is in AU for a solar-mass star.

Solution: \(P = \sqrt{4^3} = \sqrt{64} = 8\) years.

This is a standard Kepler-law shortcut that works when the star has roughly one solar mass.

3. Escape speed from the Moon

Estimate the escape speed from the Moon using \(M = 7.35 \times 10^{22}\,\mathrm{kg}\) and \(R = 1.74 \times 10^{6}\,\mathrm{m}\).

Key equation: \(v_{\rm esc} = \sqrt{2GM/R}\)

Solution: \(v_{\rm esc} = \sqrt{2(6.67 \times 10^{-11})(7.35 \times 10^{22})/(1.74 \times 10^{6})}\).

This gives \(v_{\rm esc} \approx 2.38 \times 10^{3}\,\mathrm{m\,s^{-1}}\), or about \(2.4\,\mathrm{km\,s^{-1}}\).

Magnitudes and brightness

4. Flux ratio from a magnitude difference

Star A is 5 magnitudes brighter than star B. How many times brighter is it in flux?

Key equation: \(m_2 - m_1 = -2.5\log_{10}(F_2/F_1)\)

Solution: Set \(m_B - m_A = 5\). Then

\(5 = -2.5\log_{10}(F_B/F_A)\), so \(\log_{10}(F_B/F_A) = -2\).

Therefore \(F_A/F_B = 10^2 = 100\). A 5 magnitude difference means a factor of 100 in flux.

5. Apparent and absolute magnitude

A star has apparent magnitude \(m = 8\) and is 100 pc away. Find its absolute magnitude \(M\).

Key equation: \(m - M = 5\log_{10}(d/10\,\mathrm{pc})\)

Solution: With \(d = 100\,\mathrm{pc}\),

\(m - M = 5\log_{10}(10) = 5\).

So \(8 - M = 5\), and \(M = 3\).

6. Limiting magnitude of a telescope

A telescope collects 16 times more light than the unaided eye. By how many magnitudes is its limiting brightness improved?

Key equation: \(\Delta m = 2.5\log_{10}(F_2/F_1)\)

Solution: \(\Delta m = 2.5\log_{10}(16)\).

Since \(\log_{10}(16) \approx 1.204\), \(\Delta m \approx 3.01\).

The telescope reaches about 3 magnitudes fainter than the eye, assuming equal detection conditions.

Stars and stellar physics

7. Luminosity from radius and temperature

A star has radius twice that of the Sun and surface temperature 6000 K. The Sun’s temperature is 5800 K. Estimate the luminosity ratio \(L/L_{\odot}\).

Key equation: \(L = 4\pi R^2 \sigma T^4\)

Solution: \(L/L_{\odot} = (R/R_{\odot})^2 (T/T_{\odot})^4 = 2^2 (6000/5800)^4\).

Now \((6000/5800)^4 \approx 1.14\), so \(L/L_{\odot} \approx 4.56\).

The star is about 4.6 times as luminous as the Sun.

8. Main-sequence lifetime estimate

A star has 2 solar masses. Use the rough scaling \(L \propto M^{3.5}\) and assume the main-sequence lifetime scales as \(t \propto M/L\). Estimate the lifetime relative to the Sun.

Key equation: \(t/t_{\odot} \approx (M/M_{\odot})/(L/L_{\odot})\)

Solution: With \(L/L_{\odot} \approx 2^{3.5} \approx 11.3\),

\(t/t_{\odot} \approx 2/11.3 \approx 0.18\).

If the Sun’s main-sequence lifetime is about 10 billion years, this star would live for about 1.8 billion years.

9. Surface gravity of a white dwarf

A white dwarf has the mass of the Sun and a radius of 0.01 solar radii. How does its surface gravity compare with the Sun’s?

Key equation: \(g = GM/R^2\)

Solution: The mass is the same, so

\(g_{\rm wd}/g_{\odot} = (R_{\odot}/0.01R_{\odot})^2 = 100^2 = 10^4\).

The surface gravity is about 10,000 times stronger than the Sun’s.

Cosmology and the expanding universe

10. Distance from Hubble’s law

A galaxy has recession speed 2100 km s\(^{-1}\). Estimate its distance using \(H_0 = 70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}\).

Key equation: \(v = H_0 d\)

Solution: \(d = v/H_0 = 2100/70 = 30\,\mathrm{Mpc}\).

That is about 98 million light years, using \(1\,\mathrm{Mpc} \approx 3.26\) million light years.

For a student-friendly overview of the broader topic, see the unit on The Expanding Universe and Relativistic Cosmology, the article on the Hubble tension, and how astronomers measure stellar distances.

How to study these problems

The pattern is simple. Identify the physical idea, write the key equation, convert every quantity to consistent units, and check whether the size of the answer makes sense. In olympiad papers, many questions look different on the surface, yet they often reduce to the same few relations.

• For orbits, start with Newton’s law and Kepler’s third law.
• For brightness, use the magnitude scale and the distance modulus.
• For stars, combine luminosity, radius, temperature, and mass scaling laws.
• For cosmology, begin with Hubble’s law and simple unit conversion.

Two standard references for the equations used here are the NASA overview of stars and the ESA page on measuring the Universe. For magnitude relations, the formulae are also summarised in the IAU’s guide to measuring astronomy. The basic magnitude relation appears in many textbooks and is used here in its standard form.

This kind of work fits directly into The Astronomer’s Toolkit, where you learn the units, formulae, and estimates that sit behind olympiad problems.

Frequently asked questions

Do olympiad problems usually need advanced calculus?

Most introductory olympiad questions do not. You need algebra, geometry, logarithms, and a few standard physics equations far more often than calculus.

Should I memorise every formula?

No. Learn the main equations well, then practise recognising which one applies. Many problems become much easier once you see the physical pattern.

What is the best way to check my answer?

Estimate the order of magnitude before and after the calculation. If a planet ends up with a period of hours at several AU, or a star has a tiny luminosity with a very high temperature, something went wrong.