Astronomy & Astrophysics

⭐ Key Concepts & Laws

Kepler's Third Law

$$T^2 = \frac{4\pi^2}{GM}\,a^3$$ The square of the orbital period is proportional to the cube of the semi-major axis.

Hubble's Law

$$v = H_0\,d$$ The recession velocity of a galaxy is proportional to its distance, with $H_0 \approx 70$ km/s/Mpc.

Stefan-Boltzmann Law

$$L = 4\pi R^2 \sigma T^4$$ A star's luminosity depends on its surface area and the fourth power of its temperature.

Schwarzschild Radius

$$r_s = \frac{2GM}{c^2}$$ The event horizon radius of a non-rotating black hole — the point of no return.

Nuclear Fusion (pp chain)

$$4\,{}^1\text{H} \rightarrow {}^4\text{He} + 2e^+ + 2\nu + \text{energy}$$ The proton-proton chain powers main-sequence stars like our Sun.

Drake Equation

$$N = R_* \times f_p \times n_e \times f_\ell \times f_i \times f_c \times L$$ Estimates the number of communicating civilizations in our galaxy.

Cosmic Microwave Background

$T \approx 2.725$ K blackbody radiation — the afterglow of the Big Bang, redshifted from ~3000 K over 13.8 billion years.

Special Relativity

$$E = mc^2$$ and time dilation: $t' = \dfrac{t}{\sqrt{1 - v^2/c^2}}$. Mass-energy equivalence powers the stars.

✎ Problems

1 Easy

Mars orbits the Sun at an average distance of $1.524$ AU. Using Kepler's Third Law (with Earth as reference: $T = 1$ year, $a = 1$ AU), calculate Mars's orbital period in Earth years.

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In AU-year units, Kepler's Third Law simplifies to $T^2 = a^3$. So $T = a^{3/2}$. Plug in $a = 1.524$ AU.

2 Easy

A galaxy is observed to have a redshift $z = 0.05$. Using Hubble's Law with $H_0 = 70$ km/s/Mpc, estimate the galaxy's distance. How long ago (approximately) did the light we see leave that galaxy?

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For small $z$: $v = zc$. Then $d = v/H_0$. For the lookback time, use $t \approx d/c$ (converted to years). $1$ Mpc $\approx 3.086 \times 10^{22}$ m.

3 Medium

The Sun has a surface temperature of $5{,}778$ K and luminosity of $3.846 \times 10^{26}$ W. Using the Stefan-Boltzmann Law, calculate the Sun's radius. Compare your answer to the accepted value of $6.96 \times 10^{8}$ m.

Show Hint

Rearrange $L = 4\pi R^2 \sigma T^4$ to get $R = \sqrt{\dfrac{L}{4\pi\sigma T^4}}$. Use $\sigma = 5.67 \times 10^{-8}$ W m$^{-2}$ K$^{-4}$.

4 Medium

Calculate the Schwarzschild radius for a black hole with mass equal to 10 solar masses ($M_\odot = 1.989 \times 10^{30}$ kg). What would be the average density of matter within this event horizon?

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$r_s = \dfrac{2GM}{c^2}$ with $G = 6.674 \times 10^{-11}$ N m$^2$ kg$^{-2}$. For density: $\rho = \dfrac{M}{V}$ where $V = \dfrac{4}{3}\pi r_s^3$. You may find the density is surprisingly low for supermassive black holes.

5 Medium

In the proton-proton chain, four hydrogen nuclei fuse to form one helium-4 nucleus. The mass of 4 protons is $4 \times 1.00728$ u, and He-4 is $4.00260$ u. Calculate the energy released per fusion event and estimate how many reactions per second power the Sun ($L_\odot = 3.846 \times 10^{26}$ W).

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Mass deficit: $\Delta m = 4(1.00728) - 4.00260$ u. Convert to kg ($1$ u $= 1.661 \times 10^{-27}$ kg) and use $E = \Delta m\,c^2$. Rate $= L_\odot / E$ per event.

6 Medium

A main-sequence star has twice the Sun's mass. Using the mass-luminosity relation $L \propto M^{3.5}$, how much more luminous is it than the Sun? Estimate its main-sequence lifetime relative to the Sun's (10 billion years), given that lifetime $\propto M/L$.

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$L/L_\odot = (M/M_\odot)^{3.5} = 2^{3.5}$. Lifetime ratio $= (M/M_\odot)/(L/L_\odot) = (M/M_\odot)^{-2.5}$.

7 Hard

Thought Experiment: An astronaut travels to Alpha Centauri ($4.37$ light-years away) at $0.9c$. How long does the trip take in Earth's frame? How long does it take in the astronaut's frame (proper time)? How much has each person aged when the astronaut returns?

Show Hint

Earth frame: $t = d/v$. Astronaut frame: $t' = t\sqrt{1 - v^2/c^2}$. The Lorentz factor $\gamma = \dfrac{1}{\sqrt{1 - 0.81}} \approx 2.29$. For the round trip, double both times. The twin paradox arises because only the astronaut accelerates.

8 Hard

The Cosmic Microwave Background has a temperature of $2.725$ K today. At the time of recombination ($z \approx 1100$), what was the temperature of the universe? At what wavelength does the CMB peak today, and at what wavelength did it peak at recombination?

Show Hint

Temperature scales as $T = T_0(1 + z)$. Use Wien's displacement law: $\lambda_{\max} = \dfrac{b}{T}$ where $b = 2.898 \times 10^{-3}$ m K. At recombination, the peak was in visible/near-infrared light.

9 Hard

An exoplanet orbits a Sun-like star and causes a radial velocity wobble of $50$ m/s with a period of $4$ years. The star has mass $1\,M_\odot$. Estimate the planet's minimum mass (in Jupiter masses) and its orbital radius.

Show Hint

Use Kepler's Third Law to find the orbital radius from the period. The planet's minimum mass: $M_p \sin i \approx M_* \times (v_{\text{star}}/v_p)$ where momentum conservation gives $M_p v_p = M_* v_*$. You'll need $M_p \ll M_*$ approximation.

10 Hard

Estimation Problem: Using the Drake Equation with these values: $R_* = 1.5$ stars/year, $f_p = 0.9$, $n_e = 0.4$, $f_\ell = 0.1$, $f_i = 0.1$, $f_c = 0.01$, $L = 10{,}000$ years — how many communicating civilizations might exist? Discuss which parameter you think is most uncertain.

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Simply multiply all values: $N = R_* \times f_p \times n_e \times f_\ell \times f_i \times f_c \times L$. The most debated parameters are $f_i$ (fraction developing intelligence) and $L$ (civilization lifetime), which can vary by orders of magnitude.

11 Advanced

A white dwarf has a mass of $1.2\,M_\odot$ compressed into a radius roughly equal to Earth's ($6{,}371$ km). Calculate its surface gravity, escape velocity, and the gravitational redshift of a photon emitted from its surface ($\Delta\lambda/\lambda = GM/(Rc^2)$).

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Surface gravity: $g = \dfrac{GM}{R^2}$. Escape velocity: $v_e = \sqrt{\dfrac{2GM}{R}}$. For gravitational redshift, use the weak-field approximation. Compare each value to Earth's — the ratios are dramatic.

12 Advanced

Deep Thought: The observable universe has a radius of about $46.5$ billion light-years, yet the universe is only $13.8$ billion years old. Explain this apparent paradox quantitatively. If the universe is expanding at $H_0 = 70$ km/s/Mpc, what is the current recession velocity of the most distant observable objects? Can this exceed $c$?

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The key is that space itself has been expanding while light traveled. The comoving distance exceeds $ct$ because the metric expanded. Recession velocity $v = H_0 d$ — for $d \approx 14{,}200$ Mpc, $v \approx 3.2c$. This doesn't violate special relativity because it's metric expansion, not motion through space.