Exercises: Introduction to cosmology and astrophysics

The Big Bang model is the accepted scientific framework for describing the origin and evolution of the universe. The diagram represents the expansion of the universe from the Big Bang to the present day.

1. a) Explain in your own words what this model asserts about the origin of the universe.
2. b) State the approximate age of the universe and express it in standard form, in years.
3. c) Cite two experimental observations supporting the Big Bang model.

The Hubble–Lemaître law relates the recessional velocity $v$ of a galaxy to its distance $d$ by means of $v = H_0 \cdot d$, where $H_0 = 70\ \text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$. The graph represents this law and marks the galaxy from the question.

1. a) Calculate the recessional velocity of a galaxy located $20\ \text{Mpc}$ away from us.
2. b) Explain what it means for this velocity to be positive: is the galaxy approaching or moving away?

A light year ($\text{ly}$) is the distance light travels in a vacuum in one year. Data: $c = 3\cdot10^{8}\ \text{m/s}$; $1\ \text{year} = 3,156\cdot10^{7}\ \text{s}$.

1. a) Calculate the value of $1\ \text{ly}$ in metres, in standard form and to three significant figures.
2. b) Express the result in kilometres.

Order the following distances from smallest to largest and justify the ordering using standard form:

1. a) the mean distance from Earth to the Moon, $3,84\cdot10^{8}\ \text{m}$
2. b) the astronomical unit, $1\ \text{UA} = 1,5\cdot10^{11}\ \text{m}$
3. c) the radius of Earth, $6,37\cdot10^{6}\ \text{m}$
4. d) one light year, $1\ \text{ly} = 9,46\cdot10^{15}\ \text{m}$
5. e) one parsec, $1\ \text{pc} = 3,086\cdot10^{16}\ \text{m}$

The Solar System is our nearest cosmic neighbourhood. The diagram shows its structure, with the eight unnamed planets.

1. a) Write down the eight planets in order of increasing distance from the Sun.
2. b) Classify them into rocky planets and gas giants, and state between which planets the asteroid belt lies.
3. c) What is the star of the Solar System and approximately what fraction of the total mass of the system does it contain?

Our Solar System is part of a galaxy, the Milky Way.

1. a) State what type of galaxy the Milky Way is and approximately how many stars it contains, expressed as a power of ten.
2. b) Describe where the Sun is located within the galaxy: in the centre, in an arm or at the edge?
3. c) Explain in your own words the difference between the Solar System and the Milky Way.

The ultimate fate of a star depends mainly on its initial mass.

1. a) Put the life stages of a star like the Sun in order: red giant, nebula, white dwarf, main sequence.
2. b) State what happens instead to a star much more massive than the Sun at the end of its life.
3. c) Name the two possible remnant end-states of such very massive stars.

Research in cosmology and astrophysics has driven technologies that we use every day.

1. a) The CCD sensors in mobile phone cameras were originally developed for astronomy. Cite another everyday technology related to space research or fundamental physics and explain the connection.
2. b) Explain why countries invest in telescopes and observatories even if they do not produce an immediate economic benefit.

A galaxy in the Virgo Cluster is located $150\ \text{Mpc}$ from the Milky Way. Data: $H_0 = 70\ \text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$; $c = 3\cdot10^{8}\ \text{m/s}$.

1. a) Apply the Hubble–Lemaître law, $v = H_0 \cdot d$, to calculate its recessional velocity, in $\text{km/s}$.
2. b) Express that velocity as a fraction of the speed of light and comment on whether relativistic effects would be noticeable at that speed.

Proxima Centauri is the nearest star to the Sun: it is located at $4,24\ \text{ly}$. Data: $1\ \text{ly} = 9,46\cdot10^{12}\ \text{km}$.

1. a) Express this distance in kilometres, in standard form.
2. b) How long does its light take to reach Earth?
3. c) If a spectacular flare occurred on that star today, could we observe it tonight? Reason your answer.

Sirius, the brightest star in the night sky, is located $2,64\ \text{pc}$ from the Sun. Data: $1\ \text{pc} = 3,26\ \text{ly}$; $1\ \text{ly} = 9,46\cdot10^{12}\ \text{km}$.

1. a) Express the distance to Sirius in light years.
2. b) Express it in kilometres, in standard form.
3. c) How long would a radio signal sent from Earth today take to reach Sirius?

The parsec is defined from annual parallax: a star is at $1\ \text{pc}$ if its parallax is one arcsecond. In practice, the relation $d\ (\text{pc}) = \frac{1}{p}$ is used, with the parallax $p$ measured in arcseconds. The Gaia mission measured a parallax of $0,02''$ for a star. Data: $1\ \text{pc} = 3,26\ \text{ly}$. The diagram shows the geometry of annual parallax (the figure is not to scale).

1. a) Calculate the distance to that star in parsecs.
2. b) Convert the result to light years.
3. c) Reason why this method only works for relatively nearby stars: what happens to the parallax when the star is very far away?

A simple estimate of the age of the universe is the so-called "Hubble time", $t_0 \approx \frac{1}{H_0}$. Data: $H_0 = 70\ \text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$; $1\ \text{Mpc} = 3,086\cdot10^{19}\ \text{km}$; $1\ \text{year} = 3,156\cdot10^{7}\ \text{s}$.

1. a) Express $H_0$ in units of $\text{s}^{-1}$.
2. b) Calculate $t_0 = \frac{1}{H_0}$ in seconds.
3. c) Express the result in years and compare it with the accepted age of the universe, $1,38\cdot10^{10}\ \text{years}$.

The Andromeda Galaxy (M31) is located about $2,5\cdot10^{6}\ \text{ly}$ from the Milky Way. Data: $1\ \text{pc} = 3,26\ \text{ly}$.

1. a) Express this distance in parsecs and in megaparsecs.
2. b) How long has the light with which we observe Andromeda today taken to reach us?
3. c) Explain what we mean when we say that in astronomy "looking far away is looking into the past".

The Sun radiates a power of $3,9\cdot10^{26}\ \text{W}$, obtained from the fusion of hydrogen into helium, a process in which a small part of the mass is converted into energy according to $E = m \cdot c^2$. Data: $c = 3\cdot10^{8}\ \text{m/s}$; $1\ \text{year} = 3,156\cdot10^{7}\ \text{s}$.

1. a) Calculate the mass that the Sun converts into energy each second.
2. b) Calculate the mass converted in one year.
3. c) Comment on the result: do you think the Sun is being "consumed" at a worrying rate? Why?

The redshift of the spectrum of a galaxy indicates that it is moving away from us at $2100\ \text{km/s}$. Data: $H_0 = 70\ \text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$; $1\ \text{pc} = 3,26\ \text{ly}$. The figure compares the reference spectrum, measured in the laboratory, with that observed in that galaxy.

1. a) Use the Hubble–Lemaître law to calculate how far away it is, in $\text{Mpc}$.
2. b) Express that distance in light years.
3. c) Explain what happens to the wavelength of light when the source moves away from the observer and why we say the spectrum is "redshifted".

The Voyager 1 probe, launched in 1977, is the furthest human-made object from Earth: it is located at about $160\ \text{UA}$ from the Sun and is moving away at $17\ \text{km/s}$. Data: $1\ \text{UA} = 1,5\cdot10^{8}\ \text{km}$; $1\ \text{ly} = 9,46\cdot10^{12}\ \text{km}$; $1\ \text{year} = 3,156\cdot10^{7}\ \text{s}$.

1. a) Express its current distance in kilometres and in light years.
2. b) Assuming it maintained a constant speed, calculate how many years it would take to reach a distance of $1\ \text{ly}$ from the Sun.
3. c) In light of the result, justify why crewed interstellar travel is not feasible with current technology.

Two galaxies are located $100\ \text{Mpc}$ and $300\ \text{Mpc}$ from us, respectively. Data: $H_0 = 70\ \text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$; $c = 3\cdot10^{8}\ \text{m/s}$.

1. a) Calculate the recessional velocity of each according to the Hubble–Lemaître law.
2. b) At what distance would a galaxy be located whose recessional velocity was $\frac{c}{4} = 7,5\cdot10^{4}\ \text{km/s}$?
3. c) In 1998, observations of distant supernovae showed that these galaxies are further away than predicted by a constant-rate expansion. Explain what conclusion was drawn about the expansion of the universe and what mysterious component was proposed to explain it.

It is estimated that the Sun will remain on the main sequence for about $10^{10}\ \text{years}$ in total, radiating a power of $3,9\cdot10^{26}\ \text{W}$. Data: $c = 3\cdot10^{8}\ \text{m/s}$; mass of the Sun $M_S = 1,99\cdot10^{30}\ \text{kg}$; $1\ \text{year} = 3,156\cdot10^{7}\ \text{s}$.

1. a) Calculate the total energy that the Sun will radiate throughout its main sequence stage.
2. b) Using $E = m \cdot c^2$, calculate the mass equivalent to that energy.
3. c) Find what fraction of the Sun's mass this mass represents and comment on why the Sun will not suddenly go out "for lack of fuel".
4. d) Briefly describe what will happen to the Sun from the moment it exhausts the hydrogen in its core until its final state.

The atomic clocks on GPS satellites run fast by about $38\ \mu\text{s}$ per day relative to clocks on Earth's surface, due to combined effects of special relativity and general relativity. Data: $c = 3\cdot10^{8}\ \text{m/s}$.

1. a) GPS determines position by measuring the time it takes for a radio signal, travelling at the speed of light, to arrive. Calculate the distance error resulting from a time error of $38\ \mu\text{s}$.
2. b) If relativistic effects were not corrected, estimate the position error accumulated in one day and comment on whether GPS would still be useful for navigation.
3. c) Use this example to argue why abstract theoretical physics (relativity, quantum physics) has a direct impact on everyday technology.

Current measurements of the Hubble constant do not agree with each other: the cosmic microwave background (Planck satellite) suggests $H_0 = 67\ \text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$, whereas supernovae in nearby galaxies (Hubble and James Webb telescopes) give $H_0 = 73\ \text{km}\cdot\text{s}^{-1}\cdot\text{Mpc}^{-1}$. Data: $1\ \text{Mpc} = 3,086\cdot10^{19}\ \text{km}$; $1\ \text{year} = 3,156\cdot10^{7}\ \text{s}$. The graph schematically shows the two measurements of $H_0$.

1. a) Calculate the "Hubble time", $t_0 \approx \frac{1}{H_0}$, in years, for each of the two values.
2. b) Find the difference between the two estimated ages of the universe.
3. c) This discrepancy is known as "Hubble tension". Explain why it worries cosmologists and what it might imply about the standard cosmological model.

The Milky Way has a diameter of about $10^{5}\ \text{ly}$ and the Andromeda Galaxy is located $2,5\cdot10^{6}\ \text{ly}$ from us. Data: $1\ \text{ly} = 9,46\cdot10^{15}\ \text{m}$; diameter of the Sun $d_S = 1,4\cdot10^{9}\ \text{m}$.

1. a) Calculate how many "galactic diameters" fit into the distance separating the Milky Way from Andromeda.
2. b) Build a scale model: if we were to represent the Milky Way as a plate of diameter $20\ \text{cm}$, at what distance would you place the plate representing Andromeda?
3. c) Calculate what size the Sun would be in that same model, express it in standard form and comment on what you conclude about how "empty" the universe is.

A black hole is the ultimate remnant of very massive stars. Its "boundary" is the event horizon, whose radius (Schwarzschild radius) is given by $R_S = \frac{2 \cdot G \cdot M}{c^2}$. Data: $G = 6,67\cdot10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$; $c = 3\cdot10^{8}\ \text{m/s}$; mass of the Sun $M_S = 1,99\cdot10^{30}\ \text{kg}$.

1. a) Calculate the Schwarzschild radius of the Sun, that is, the size to which it would have to be compressed to become a black hole.
2. b) Without repeating the whole calculation, reasoning by proportionality, find the Schwarzschild radius of a stellar remnant of $25\ M_S$.
3. c) Explain what stages a star of $25\ M_S$ goes through from the main sequence to forming that black hole.

A group of students analyses data from three galaxies obtained using a school telescope:

• galaxy 1: distance $10\ \text{Mpc}$, recessional velocity $700\ \text{km/s}$

• galaxy 2: distance $25\ \text{Mpc}$, recessional velocity $1750\ \text{km/s}$

• galaxy 3: distance $40\ \text{Mpc}$, recessional velocity $2800\ \text{km/s}$

The graph represents the data for the three galaxies.

1. a) Calculate the ratio $\frac{v}{d}$ for each galaxy and check that it is approximately constant.
2. b) Identify what quantity this ratio represents and give its value with its units.
3. c) Predict the recessional velocity of a fourth galaxy located $60\ \text{Mpc}$ away.
4. d) Explain why this proportionality is evidence for the expansion of the universe and not for galaxies "fleeing" from us by moving through space.

You can do it on paper and send a photo with your answer.

Unit challenge. How do we measure the universe if we cannot travel through it? Become an astrophysicist and reproduce the foundation of "standard candles" to understand cosmic expansion.

1. Home measurement: In a dark room, place a point light bulb and measure the illuminance $E$ at 5 different distances $r$ (between $0,20\text{ m}$ and $1,00\text{ m}$) using the free app Phyphox (your phone's light sensor). Record the associated instrumental uncertainties ($\pm\Delta r$, $\pm\Delta E$).

2. Data processing: Prepare a table with your values ($r$, $1/r^2$, $E$). Plot $E$ versus $1/r^2$, draw the best-fit line, and calculate its slope with its uncertainty to test the inverse-square law of luminous flux.

3. Cosmological connection: Look up distance and brightness data for two real Type Ia supernovae on the official ESA or NASA website. Explain why knowing their luminosity allows us to measure extragalactic distances and support the Hubble-Lemaître law.

4. Final product: Submit a brief report (two pages) integrating the table, the analyzed graph, and your cosmological justification.

Grading criteria:

• Experimental rigor in taking home measurements and handling uncertainties.

• Technical quality of the table and the graphical representation with appropriate linearization.

• Coherence in the physical justification of standard candles in modern cosmology.