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Universal constants Astronomical data Calculus related formulas Условие Решение General Marking Principles

IOAA 2025, Теория — Задача 10. Big Bang Nucleosynthesis

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Всички материали от IOAA 2025

IOAA 2025 — Theory Examination · 35 т.

Theory Examination Instructions

You are given an envelope marked B. Please verify the Contestant Code printed on it. Inside you will find:

This Instruction Sheet and/or This Instruction Sheet in your native language (if applicable)
Data Sheet and/or Data Sheet in your native language (if applicable)
A set of Working Sheets
One envelope marked S (please verify the Contestant Code marked on it).

You are allowed to open the envelope S only when the whistle is blown to mark the start of the examination.

Inside the envelope S you will find the following:

Question Paper and/or Question Paper in your native language (if applicable)
A set of Summary Answersheets.

Total duration of this examination is 5 hours.

This examination consists of the following questions:

Questions	Marks each	Subtotal
T01 and T02	10	$2\times10=20$ marks
T03 and T04	15	$2\times15=30$ marks
T05–T08	20	$4\times20=80$ marks
T09 and T10	35	$2\times35=70$ marks
T11 and T12	50	$2\times50=100$ marks
Total		300 marks

Writing your answers:

Summary Answersheet:

For each question a separate Summary Answersheet has been provided. Write your contestant code on each page of the Summary Answersheet.
Final answer(s) for each question / each part of a question must be written in the corresponding box in the Summary Answersheet.

Working Sheets:

Your detailed work must be shown on the Working Sheet.
Write your Contestant Code on both sides of each Working Sheet.
Start a new question on a fresh Working Sheet. Write the question number in the box provided at the top.
Both sides of the Working Sheet should be used. Write only inside the boxed area.
For rough work that you do not want to be evaluated, clearly strike that part out.
For each question write the page number in the box provided at the top of the Working Sheet, starting with 1 (each sheet has 2 pages). For example, if you use only 1 Working Sheet for T01, the question number and page number should be “T01, 1” and “T01, 2” on the FRONT and BACK of that Working Sheet, respectively. Next, if you use 2 Working Sheets for T02, the question number and page number should be “T02, 1” and “T02, 2” for the first Working Sheet, and “T02, 3” and “T02, 4” on the second Working Sheet.
You may ask for extra Working Sheets, if required.
Note that failure to mark the Working Sheets, as instructed above, may result in your answers not being evaluated.

At the end of the examination:

Check that you have correctly written your Contestant Code and page numbers on all used sheets.
Arrange your sheets in the following way: Summary Answersheet of each question (starting with T01) followed by Working Sheet(s) used for that question.
Similarly stack the Summary Answersheet and Working Sheet(s) used for each question, in sequence.
Place the bunch of all your used sheets arranged in the above manner inside the smaller envelope S. All Summary Answersheets must be placed inside the envelope S irrespective of whether you have used them or not.
Only the contents of the smaller envelope S will be evaluated. Any sheet that is not placed inside this envelope will not be evaluated.
Place the smaller envelope S inside the bigger envelope B.
All other sheets, namely, Instruction Sheet(s), Question Paper(s), Data Sheet(s) and unused Working Sheet(s) should be placed directly inside the bigger envelope B.
Tie the thread seal on the envelope B.
Hand over the envelope B to the examiner. Make sure that no sheet is left on the table, except the green and the red card.

General Instructions:

If you need a visit to the toilet during the examination, please raise your hand holding the green card kept on your table. You will be escorted by a volunteer.
If you need any other help during the examination, please raise your hand holding the red card kept on your table.
Use only black or blue pen for writing. For drawing, you may use a pencil.
Some marks may be deducted if the final answer is given without detailed solution.
Some marks may be deducted if the final answers have improper number of significant digits, no units or wrong units.
The necessary values of fundamental and astronomical constants should be taken from the Data Sheet provided to you.

Original IOAA 2025 Data Sheet

Universal constants

Quantity	Symbol	Value
Avogadro constant	$N_A$	$6.022\times10^{23}\ \mathrm{mol^{-1}}$
Boltzmann constant	$k_B$	$1.381\times10^{-23}\ \mathrm{J\,K^{-1}}$
Charge of electron	$e$	$1.602\times10^{-19}\ \mathrm C$
Planck constant	$h$	$6.626\times10^{-34}\ \mathrm{J\,s}$
Speed of light in vacuum	$c$	$2.998\times10^8\ \mathrm{m\,s^{-1}}$
Universal gravitational constant	$G$	$6.674\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}$
Universal gas constant	$R$	$8.315\ \mathrm{J\,mol^{-1}\,K^{-1}}$
Stefan-Boltzmann constant	$\sigma$	$5.670\times10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}}$
Wien’s displacement constant	$b$	$2.898\times10^{-3}\ \mathrm{m\,K}$
Permittivity of free space	$\epsilon_0$	$8.854\times10^{-12}\ \mathrm{m^{-3}\,kg^{-1}\,s^4\,A^2}$
Permeability of free space	$\mu_0$	$1.257\times10^{-6}\ \mathrm{N\,A^{-2}}$
Mass of electron	$m_e$	$9.109\times10^{-31}\ \mathrm{kg}=0.511\ \mathrm{MeV}/c^2$
Mass of proton	$m_p$	$1.673\times10^{-27}\ \mathrm{kg}=938.272\ \mathrm{MeV}/c^2$
Mass of neutron	$m_n$	$1.675\times10^{-27}\ \mathrm{kg}=939.565\ \mathrm{MeV}/c^2$
Mass of deuteron	$m_D$	$3.344\times10^{-27}\ \mathrm{kg}=1875.613\ \mathrm{MeV}/c^2$
Mass of He nucleus	$m_{\mathrm{He}}$	$6.645\times10^{-27}\ \mathrm{kg}=3727.181\ \mathrm{MeV}/c^2$

Astronomical data

Quantity	Symbol	Value
Mass of Sun	$M_\odot$	$1.988\times10^{30}\ \mathrm{kg}$
Radius of Sun	$R_\odot$	$6.957\times10^8\ \mathrm m$
Luminosity of Sun	$L_\odot$	$3.828\times10^{26}\ \mathrm W$
Effective temperature of Sun	$T_{\mathrm{eff},\odot}$	$5772\ \mathrm K$
Apparent magnitude of Sun (in V-band)	$m_{V,\odot}$	$-26.74$
Absolute magnitude of Sun (in V-band)	$M_{V,\odot}$	$+4.82$
Apparent bolometric magnitude of Sun	$m_{\mathrm{bol},\odot}$	$-26.83$
Absolute bolometric magnitude of Sun	$M_{\mathrm{bol},\odot}$	$+4.74$
Solar constant (above atmosphere of Earth)	$S_\odot$	$1361\ \mathrm{W\,m^{-2}}$
Apparent angular diameter of Sun (from Earth)	$\theta_\odot$	$\approx32'$
Mass of Earth	$M_\oplus$	$5.972\times10^{24}\ \mathrm{kg}$
Radius of Earth	$R_\oplus$	$6.378\times10^6\ \mathrm m$
Axial tilt of Earth	$\epsilon$	$23^\circ26'$
Inclination of the lunar orbit w.r.t. the ecliptic		$5^\circ8'43''$
Mass of Jupiter	$M_J$	$1.898\times10^{27}\ \mathrm{kg}$
Radius of Jupiter	$R_J$	$6.991\times10^7\ \mathrm m$
1 Astronomical Unit	1 au	$1.496\times10^{11}\ \mathrm m$
1 parsec	1 pc	$3.086\times10^{16}\ \mathrm m$
1 light-year	1 ly	$9.461\times10^{15}\ \mathrm m$
1 jansky	1 Jy	$10^{-26}\ \mathrm{W\,m^{-2}\,Hz^{-1}}$
1 tropical year		$365.2422$ solar days $=3.156\times10^7\ \mathrm s$ ; $365\ \mathrm d\ 5\ \mathrm h\ 48\ \mathrm{min}\ 46\ \mathrm s$
1 sidereal year		$365.2564$ solar days $=3.156\times10^7\ \mathrm s$ ; $365\ \mathrm d\ 6\ \mathrm h\ 9\ \mathrm{min}\ 13\ \mathrm s$
Rate of precession of Vernal Equinox		$1^\circ$ per $71.6$ years

Calculus related formulas

$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$ .
$\frac{d}{dx}x^n=nx^{n-1}$ .
$\frac{d}{dx}\sin kx=k\cos kx$ .
$\frac{d}{dx}\cos kx=-k\sin kx$ .
$\frac{d}{dx}\tan kx=k\sec^2kx$ .
$\int x^n\,dx=\frac{x^{n+1}}{n+1}+\text{constant}$ ; for $n\ne-1$ .
$f(x)\simeq f(x_0)+\left.\frac{df}{dx}\right|_{x=x_0}(x-x_0)$ ; for $x\approx x_0$ .

Условие

During the radiation dominated era in the early Universe, the scale factor of the Universe $a\propto t^{1/2}$ , where $t$ is the time since Big Bang. During most of this era, neutrons (n) and protons (p) remain in thermal equilibrium with each other via weak interactions. The number density ( $N$ ) of free neutrons or protons is related to the temperature $T$ and their corresponding masses $m$ such that

N\propto m^{3/2}\exp\left(-\frac{mc^2}{k_BT}\right),

as long as time $t<t_\mathrm{wk}=1.70\,\mathrm{s}$ , when $k_BT>k_BT_\mathrm{wk}=800\,\mathrm{keV}$ . After $t_\mathrm{wk}$ , the weak interactions can no longer maintain such equilibrium, and free neutrons decay to protons with a half-life time of $610.4\,\mathrm{s}$ .

(T10.1) Let the number density of protons be $N_p$ , and that of neutrons be $N_n$ . Calculate the relative abundance of neutrons given the ratio $X_{n,\mathrm{wk}}=N_n/(N_n+N_p)$ at time $t_\mathrm{wk}$ . [4 т.]

(T10.2) Photons maintain thermal equilibrium and retain a blackbody spectrum at all epochs.

(T10.2a) Find the index $\beta$ , such that $T(a)\propto a^\beta$ . [2 т.]

(T10.2b) Identify which of the following graphs shows the correct behaviour of the spectral energy density for two temperatures $T_1$ and $T_2$ . Tick (✓) the correct option in the Summary Answersheet.

All five original blackbody options A–E, both coloured curves and all spectral axes/legends.

[2 т.]

(T10.3) After $t_\mathrm{wk}$ , the process of formation of deuterium from protons and neutrons is governed by the Saha equation, given by the Indian physicist Prof. Meghnad Saha, which can be simplified to

\frac{N_D}{N_n}=6.5\eta\left(\frac{k_BT}{m_nc^2}\right)^{3/2}\exp\left[-\frac{(m_D-m_p-m_n)c^2}{k_BT}\right].

Here, baryon-to-photon ratio $\eta$ is $6.1\times10^{-10}$ , and $N_D$ is the number density of deuterium.

(T10.3a) Plot the ratio $N_D/N_n$ on the grid in the Summary Answersheet, for at least 4 reasonably spaced values of temperature that lie in the domain $k_BT=[60,70]\,\mathrm{keV}$ , and draw a smooth curve passing through these points. [5 т.]

(T10.3b) From the plot find $k_BT_\mathrm{nuc}$ (in keV) when $N_D=N_n$ . [1 т.]

(T10.3c) Instead, now assume that all the free neutrons combine instantaneously with the protons at $k_BT_\mathrm{nuc}$ to form Deuterium, and all of which immediately gets converted to Helium ( ${}^4_2\mathrm{He}$ ). Compute the corresponding epoch or time of nucleosynthesis, $t_\mathrm{nuc}$ (in s), for the formation of Helium. [4 т.]

(T10.4) Calculate the value of $X_{n,\mathrm{nuc}}$ immediately before $t_\mathrm{nuc}$ . [5 т.]

(T10.5) The primordial Helium abundance, $Y_\mathrm{prim}$ , is defined to be the fraction of total mass in the Universe that is bound in Helium at $T_\mathrm{nuc}$ . Obtain a theoretical estimate for the value of $Y_\mathrm{prim}$ . For the purpose of this calculation alone, assume $m_p\approx m_n$ and that the mass of Helium, $m_\mathrm{He}\approx4m_n$ . [3 т.]

(T10.6) The primordial abundance of Helium is very difficult to measure, as stars continuously convert Hydrogen to Helium in the Universe. The amount of processing by stars in a galaxy is characterised by the relative number density of Oxygen (which is only produced by stars) to hydrogen, denoted as (O/H), in the galaxy. A compilation of the measurements of (O/H) and the Helium abundance, $Y$ , for different galaxies is plotted below.

Complete original helium-versus-oxygen scatter plot, every point/error bar, all axes and numeric ticks.

Use all of the points in this plot (which is reproduced in the Summary Answersheet) to answer the following.

(T10.6a) Estimate $Y$ for a blue compact dwarf galaxy with a value of $(O/H)=1.75\times10^{-4}$ . [2 т.]

(T10.6b) Obtain the slope $dY/d(O/H)$ of the straight line fit to the above data. [2 т.]

(T10.6c) Estimate the primordial Helium abundance, $Y_\mathrm{prim}^\mathrm{obs}$ , based on the above observations. [2 т.]

(T10.7) The deviation between $Y_\mathrm{prim}$ and $Y_\mathrm{prim}^\mathrm{obs}$ can be reconciled by changing the baryon-to-photon ratio $\eta$ . When $\eta$ is decreased, as indicated by ↓ in the Summary Answersheet, indicate the increase (↑) or decrease (↓) in $N_D/N_n(T)$ , $T_\mathrm{nuc}$ (when $N_D=N_n$ ), $X_{n,\mathrm{nuc}}$ , and $Y_\mathrm{prim}$ in the boxes provided in the Summary Answersheet. [3 т.]

Complete original Summary Answersheet T10, SA-19, FRONT: all native blank fields, printed markings, contestant-code boxes, graphs/checkboxes and page edges retained. No field filled.

Complete original Summary Answersheet T10, SA-20, BACK: all native blank fields, printed markings, contestant-code boxes, graphs/checkboxes and page edges retained. No field filled.

Решение

N\propto m^{3/2}\exp\left(-\frac{mc^2}{k_BT}\right),

as long as time $t\le t_\mathrm{wk}=1.70\,\mathrm{s}$ , when $k_BT\ge k_BT_\mathrm{wk}=800\,\mathrm{keV}$ . After $t_\mathrm{wk}$ , the weak interactions can no longer maintain such equilibrium, and free neutrons decay to protons with a half-life time of 610.4 s.

(T10.1)

The ratio of the number density of protons to that of neutrons at $t_\mathrm{wk}$ will be given by

\frac{N_n(t_\mathrm{wk})}{N_p(t_\mathrm{wk})}=\left(\frac{m_n}{m_p}\right)^{3/2}\exp\left(-\frac{(m_n-m_p)c^2}{k_BT_\mathrm{wk}}\right)\qquad\text{(2.0)}.

Substituting the values of the masses of neutron and proton, and $k_BT_\mathrm{wk}=800\,\mathrm{keV}$ , one obtains $\boxed{X_n(t_\mathrm{wk})=\frac{N_n}{N_p+N_n}=0.166}$ (2.0).

(T10.2a)

For a black body spectrum, $\lambda_\mathrm{max}T$ should be equal to a constant (1.0). Since the wavelength of light is $\lambda\propto a$ , this implies $T\propto a^{-1}$ , which implies $\boxed{\beta=-1}$ (1.0).

Alternatively, for black body radiation, energy density is proportional to $T^4$ , but the energy density is proportional to $a^{-4}$ . Therefore, $T\propto a^{-1}$ .

(T10.2b)

Graph B should be ticked. Wien’s law removes options D and E, Planck curves at different temperatures should not intersect, and so C and A are also incorrect.

(T10.3a)

Complete original hand-plotted Saha curve, four circled points, logarithmic grid and all axis labels.

	Marks
Each of the 4 points correctly located on the graph	1.0
Drawing a smooth curve passing through all the points (within one grid point)	1.0

Non-smooth lines, or straight lines not passing through all points get 0 credit.

(T10.3b)

The value of $\boxed{k_BT_\mathrm{nuc}=66.0\,\mathrm{keV}}$ (1.0).

Half credit lower limit	Full credit range	Half credit upper limit
	$65.4\,\mathrm{keV}$ to $66.6\,\mathrm{keV}$

(T10.3c)

As $T\propto a^{-1}$ , and $a\propto t^{1/2}$ , we have $T\propto t^{-1/2}$ (2.0). The temperature and time when all neutrons and protons fuse into Helium are $T_\mathrm{nuc}$ (66.0 keV) and $t_\mathrm{nuc}$ , respectively.

\boxed{t_\mathrm{nuc}=t_\mathrm{wk}\left(\frac{T_\mathrm{wk}}{T_\mathrm{nuc}}\right)^2=250\,\mathrm{s}}\qquad\text{(2.0)}.

(T10.4)

During $t_\mathrm{wk}<t<t_\mathrm{nuc}$ , as the Universe cools from 800 keV to 66.0 keV, neutrons decay into protons and with a half life of 610.4 s, which corresponds to a mean life time $\tau=880.6\,\mathrm{s}$ (1.0).

The quantity $(N_p+N_n)a^3$ does not change as the increase in protons is compensated by the decrease in neutrons (1.0).

However, $N_na^3$ will go down by an exponential factor compared to its value computed in the first question as the neutrons decay into protons, i.e.,

X_{n,\mathrm{nuc}}=X_{n,\mathrm{wk}}\exp(-(t_\mathrm{nuc}-t_\mathrm{wk}/880.6))\qquad\text{(2.0)},

\boxed{X_{n,\mathrm{nuc}}=X_{n,\mathrm{wk}}\exp(-(250-1.70)/880.6)=0.125}\qquad\text{(1.0)}.

Give part credit of 1 marks if half life time is used instead of the mean life time in the denominator of the exponential.

(T10.5)

Helium forms with the combination of 2 protons and 2 neutrons (ignoring mass deficit), and thus a number density $N_n$ of neutrons and correspondingly same number density of protons will be locked in Helium. The number density of both species goes as $a^{-3}$ , and so will cancel from the numerator and denominator (1.0).

Even after Helium is formed the sum of the mass of Helium and Hydrogen will still be the mass of all the neutrons and protons (1.0).

Thus, the abundance of Helium by mass should be

\boxed{Y_\mathrm{prim}=\frac{(m_p+m_n)N_n}{m_pN_p+m_nN_n}\approx2X_{n,\mathrm{nuc}}=0.250}\qquad\text{(1.0)}.

Alternatively, the value of $X_{n,\mathrm{nuc}}$ is $1/8$ . After formation of Helium all neutrons will combine with protons. Therefore, the mass in Helium will be $2/8$ of the total mass.

(T10.6a)

By fitting a line to these data points the Helium abundance for the said galaxy would be: $\boxed{Y=0.2480}$ (2.0).

Half credit lower limit	Full credit range	Half credit upper limit
0.2470	0.2475 to 0.2485	0.2490

Complete original helium/oxygen fitted graph, all points/error bars and handwritten slope/intercept annotations.

(T10.6b)

The slope of the line yields $\boxed{dY/d(O/H)=23}$ (2.0).

Half credit lower limit	Full credit range	Half credit upper limit
15	19 to 27	31

(T10.6c)

The primordial Helium abundance, $\boxed{Y_\mathrm{prim}^\mathrm{obs}=0.2440}$ (2.0).

Half credit lower limit	Full credit range	Half credit upper limit
0.2430	0.2435 to 0.2445	0.2450

(T10.7)

From the Saha equation, we see that $N_D/N_n(T)\propto\eta$ , thus when $\eta\downarrow$ , $N_D/N_n(T)$ will also $\downarrow$ . From the plot in (T10.3a), this would mean that $N_D/N_n(T)$ would be unity at a lower value of $T_\mathrm{nuc}$ which will go $\downarrow$ . A lower temperature corresponds to a longer $t_\mathrm{nuc}$ ( $\uparrow$ ). This would result in more neutrons decaying into protons and thus less $Y_\mathrm{prim}\downarrow$ .

Quantity	Change
$\eta$	↓
$N_D/N_n(T)$	↓
$T_\mathrm{nuc}$ ( $N_D=N_n$ )	↓
$t_\mathrm{nuc}$	↑
$X_{n,\mathrm{nuc}}$	↓
$Y_\mathrm{prim}$	↓

Each correct box fetches 0.5 marks. We give a bonus 0.5 marks if all 5 boxes are marked correctly.

General Marking Principles

STRICT adherence to given marking scheme.
No double penalty – deduction only at the original place of error; no further penalty in consequent steps, if they are otherwise correct – if final answer is physically unviable or dimensionally incorrect, deduct for final answer also.
Significant figures: one more or one less than appropriate no. of significant figures allowed with full credit.
Units / dimensions:
- In cases where units are not mentioned in the Summary Answersheet, 0.5 deduction for missing units.
- For missing units on the same kind of quantities in a given part, deduct only once.
- For wrong dimensions in final answer because of some error in earlier steps, no credit for final answer, irrespective of no-double penalty principle.
If expression is asked in terms of x, y, …: deduction of final part credit if expression is not in terms of appropriate quantities.
Cumulative part marking: all previous part credits to be added if final expression / number is correct (unless otherwise specified).
Final answer not in Summary Answersheet, but present inside: if final answer is present on Working Sheet, but not transferred on Summary Answersheet, it is to be credited.

Бележки за изданието

The printed question, key and blank answer sheets contain differing expressions and units. The Balmer key uses different rounded colour excesses; the quasar scale changes its angular unit; the neutron-star key prints differing mass-rate units/values. The black-hole-shadow question prints an unsquared radial speed in its comparison equation, while the repeated key uses a square. Atmospheric Seeing asks for a diameter but its key describes a radius. The nucleosynthesis key changes strict inequalities, has a differently grouped decay exponent, and includes an extra time-arrow field also present on its answer sheet. The stellar-profile key omits a factor in one line that appears in the next. The Hawking key prints a separation step differing from its integral and a horizon volume differing from its prose radius. Its answer sheet labels a temperature field with “s”. These native forms are preserved, including every original blank answer page.

Оригинал в Архива: Theory-Questions.pdf · официални решения: Theory-Solution.pdf · допълнителни данни: Theory-Summary-Answersheet.pdf · допълнителни данни: Theory-Instructions.pdf · допълнителни данни: Data_Sheet.pdf · допълнителни данни: IOAA2025_General_Marking_Scheme.pdf