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

IOAA 2025, Теория — Задача 5. Quasars

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Всички материали от IOAA 2025

IOAA 2025 — Theory Examination · 20 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

A quasar is an extremely luminous active galaxy powered by a supermassive black hole that emits relativistic jets. The figure shows a series of panels of radio images of a quasar (with redshift $z=0.53$ , and luminosity distance $D_L=1.00\times10^{10}\,\mathrm{ly}$ ) at different times. The “core” aligns with the vertical white line, while a jet, consisting of a “blob” (marked white $+$ ), moves away from it over time. Each panel shows the observation time (starting with $T_0$ for the first image), and the angular scale is indicated at the top and bottom of the figure.

All five original coloured quasar radio panels, time labels and full angular tick scales.

(T05.1) Determine the blob’s angular separation, $\phi_{\mathrm{blob}}$ (in milliarcsecond), and its transverse distance, $l_{\mathrm{blob}}$ (in light-year), from the quasar core for each observation. Then, calculate the blob’s apparent velocity in the transverse direction ( $v_{\mathrm{app}}$ ) as a fraction of light speed, $\beta_{\mathrm{app}}$ ( $=v_{\mathrm{app}}/c$ ) by using consecutive observations. Also calculate the average apparent velocity $\beta_{\mathrm{app}}^{\mathrm{ave}}$ over the entire observation period. [5 т.]

The quasar jet actually moves at a relativistic speed $v=\beta c$ , but not necessarily in the plane of the sky: e.g., it makes an angle $\theta$ (the “viewing angle”) with respect to the line of sight of a distant observer (indicated by the dashed lines), as shown in the sketch below.

For this and all subsequent parts, ignore redshift of the quasar and any relativistic effects.

Original quasar Core/A/B geometry with all line-of-sight, angle and observer-time panels.

(T05.2) The light emitted by the blob at two different times $t_0$ (corresponding to position A) and $t_0+\Delta t$ (corresponding to position B) reaches the observer at $t_A$ and $t_B$ , respectively. Thus the observed time difference is $\Delta t_{\mathrm{app}}=t_B-t_A$ .

(T05.2a) Find an expression for the ratio $\dfrac{\Delta t_{\mathrm{app}}}{\Delta t}$ in terms of $\beta$ and $\theta$ . [2 т.]

(T05.2b) Using this ratio, express $\beta_{\mathrm{app}}$ in terms of $\beta$ and $\theta$ . [2 т.]

(T05.3) Motion is called superluminal if the apparent speed exceeds that of light ( $\beta_{\mathrm{app}}>1$ ), and subluminal if it does not ( $\beta_{\mathrm{app}}<1$ ).

(T05.3a) For $\beta_{\mathrm{app}}=1$ , plot a smooth curve of $\beta$ as a function of $\theta$ to mark the boundary between subluminal and superluminal motions. Shade the superluminal region in the graph with slanted lines (////). [4 т.]

(T05.3b) Find the lowest true jet speed ( $\beta_{\mathrm{low}}=v_{\mathrm{low}}/c$ ) for the superluminal motion to occur and also its corresponding viewing angle $\theta_{\mathrm{low}}$ . [2 т.]

(T05.4) Find an expression for the maximum viewing angle, $\theta_{\max}$ , for which a given value of $\beta_{\mathrm{app}}$ will be possible. [2 т.]

The core of a quasar, its central compact object, exhibits variability in its emission due to internal processes occurring within a causally connected region. The size (= radius) of this region is typically taken to be about five times the Schwarzschild radius of the core.

(T05.5) The core of a certain quasar is found to vary on time scales of about 1 h. Obtain an upper limit, $M_{c,\max}$ , on the mass of the central compact object, in units of solar mass. [3 т.]

Complete original Summary Answersheet T05, SA-9, FRONT: all native blank fields, printed markings, contestant-code boxes, graphs/checkboxes and page edges retained. No field filled.

Complete original Summary Answersheet T05, SA-10, BACK: all native blank fields, printed markings, contestant-code boxes, graphs/checkboxes and page edges retained. No field filled.

Решение

(T05.1)

Angular diameter distance $=D_L/(1+z)^2$ .

\text{Scale}=\text{Angular diameter distance}\times\frac\pi{60\times60\times180}\,\mathrm{ly/arcsecond}

=20.7\,\mathrm{ly\,milliarcsecond^{-1}}\qquad\text{(1.0)}.

Date of observation	$\phi_\mathrm{blob}$ (milliarcsecond)	$l_\mathrm{blob}$ (ly)	$\beta_\mathrm{app}$
$T_0$	1.6	33	×
$T_0+605\,\mathrm{d}$	2.1	44	6.6
$T_0+1186\,\mathrm{d}$	2.3	48	2.5
$T_0+1786\,\mathrm{d}$	2.8	58	6.1
$T_0+2338\,\mathrm{d}$	3.1	64	4.0

(3.0)

	Marks
All $\phi_\mathrm{blob}$ entries are correct	1.0
3–4 $\phi_\mathrm{blob}$ entries are correct	0.5

Similar marking for $l_\mathrm{blob}$ and $\beta_\mathrm{app}$ , but without double penalty. Tolerance in each $\phi_\mathrm{blob}$ is $\pm0.1$ milliarcsecond. Tolerance in each $l_\mathrm{blob}$ is $\pm2\,\mathrm{ly}$ . Tolerance in each $\beta_\mathrm{app}$ is $\pm1.2$ .

The average apparent velocity can be calculated by using the distance and time between the first and the last point which yields $\boxed{\beta_\mathrm{app}^\mathrm{ave}=4.8}$ .

Half credit lower limit	Full credit range	Half credit upper limit
	3.6 to 6.4

(T05.2a)

Radiation from the core always takes the same time to reach us, but the time taken by radiation from the moving blob in the jet decreases since its distance from us reduces.

Original handwritten quasar component geometry, all projection labels and observer-time panels.

As shown in the sketch above, a signal emitted from the moving blob at B covers a distance $(\beta c)\Delta t\cos\theta$ less than a signal emitted at A. Thus the apparent time difference between the light received by the observer is

\Delta t_\mathrm{app}=t_B-t_A=\left[t_0+\Delta t-\frac{(\beta c)\Delta t\cos\theta}c\right]-t_0=\Delta t(1-\beta\cos\theta),

\implies\boxed{\frac{\Delta t_\mathrm{app}}{\Delta t}=(1-\beta\cos\theta)}\qquad\text{(2.0)}.

In case the answer is incorrect but the diagram has the correct entries for the sides of the relevant triangle (i.e., components of $\beta c\Delta t$ ), then we give 1 mark as partial credit.

(T05.2b)

The apparent velocity on the sky is

\beta_\mathrm{app}=\frac{v_\mathrm{app}}c=\frac{\beta\Delta t\sin\theta}{\Delta t_\mathrm{app}},

\boxed{\beta_\mathrm{app}=\frac{\beta\sin\theta}{1-\beta\cos\theta}}\qquad\text{(2.0)}.

The resulting formula indicates that the measured apparent velocity of the jet’s component is increased for relativistic features and can be higher than speed of light for small values of angle $\theta$ .

(T05.3a)

For $\beta_\mathrm{app}=1$ , we obtain

\beta(\sin\theta+\cos\theta)=1\implies\beta=\frac1{\sin\theta+\cos\theta}.

Superluminal motion occurs if $\beta_\mathrm{app}>1$ ; then

\beta(\sin\theta+\cos\theta)>1.

This defines the superluminal region which is to be shaded.

Original full superluminal-speed curve, native plotted points, hatching, axes and complete grid.

	Marks
Correct values at each of $15^\circ$ , $30^\circ$ , $45^\circ$ , $60^\circ$ and $75^\circ$	2.5
Curve is overall smooth and symmetric	0.5
Correct region is shaded	1.0

(T05.3b)

It may be seen from graph drawn in (T05.3a) that the minimum value of $\beta$ for which superluminal motion is apparent occurs at $\theta=45^\circ$ , i.e., $\beta=1/\sqrt2$ .

\boxed{\beta_\mathrm{low}=1/\sqrt2\approx0.707}\qquad\text{(1.0)},

and the corresponding orientation angle, $\boxed{\theta_\mathrm{low}=45^\circ}$ (1.0).

(T05.4)

We know

\beta_\mathrm{app}=\frac{\beta\sin\theta}{1-\beta\cos\theta}.

Thus

\beta\left(\frac{\sin\theta+\beta_\mathrm{app}\cos\theta}{\beta_\mathrm{app}}\right)=1.

Since $\beta\le1$ , we have

\frac{\sin\theta+\beta_\mathrm{app}\cos\theta}{\beta_\mathrm{app}}>1\qquad\text{(1.0)},

or $\beta_\mathrm{app}\tan(\theta/2)<1$ . Therefore, the maximum viewing angle for a jet with a given $\beta_\mathrm{app}$ is

\boxed{\theta_\mathrm{max}=2\tan^{-1}\left[\frac1{\beta_\mathrm{app}}\right]}\qquad\text{(1.0)}.

(T05.5)

Time-scale $\delta t$ of any substantial variability from a source cannot be shorter than the light crossing time of the source. This then gives an upper limit to the size $\lesssim c\delta t$ , where $c$ is the speed of light. Therefore,

c\delta t\gtrsim5R_\mathrm{SC}=5\times\frac{2GM}{c^2}\qquad\text{(1.0)},

where, $M$ is the mass of the central compact object. Rearranging,

M\lesssim\frac{c^3\delta t}{10G}.

Thus, upper limit on the mass of central compact core of quasar

M_{c,\mathrm{max}}=\frac{c^3\delta t}{10G}\qquad\text{(1.0)}

=\frac{(2.998\times10^8)^3\times3600}{10\times6.674\times10^{-11}}\,\mathrm{kg}=7.311\times10^7M_\odot,

\boxed{M_{c,\mathrm{max}}=7\times10^7M_\odot}\qquad\text{(1.0)}.

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