Съдържание

IOAA 2016, Теория, английски — Задача 12. AstroSat

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Всички материали от IOAA 2016

Theoretical Examination · 50 т.

Внимание: Бележка към темата

Both archived English question editions print T12 AstroSat, but both 27-page English worked keys substitute T12 Exoplanets. No matching AstroSat solution was found in the inspected full year inventory. Summary Answer Sheets and Cover Sheet named by the instructions were not found in this archived source pack. Instructions specify 13 questions totalling 300 points; both question editions contain 12 questions totalling 250 points. Printed numerical/formula discrepancies are retained: T8 uses c/lambda_max for nu_max and prints 1233 pi then 3813; T9 numerical velocity denominator lacks a power of ten and the stated semimajor axis is 20933 km; T11.4 prints a direct rather than inverse mass ratio in the radius-cubed relation; T11.7 prose says 166.67 Hz while its mass calculation uses 142.86 Hz. No inferred corrections are substituted.

Instructions to the Contestants

(1) Total time duration of this examination is five hours.

(2) This examination consists of

Questions	Marks each	Total
Question 1 to 5	10	10 x 5 = 50 marks
Question 6 to 10	20	20 x 5 = 100 marks
Question 11 to 13	50	50 x 3 = 150 marks
	Total	300 marks

(3) Inside the envelope you will find

a. Question paper in English

b. Question paper in your native language (if applicable)

c. Table of Constants in English

d. Table of Constants in your native language (if applicable)

e. A set of Summary Answersheets

f. Cover Sheet

(4) Use only black or blue pen for writing. For figures, you may use pencils.

(5) There is no negative marking.

(6) Some marks will be deducted if the final answer is given without detailed solution.

(7) Some marks will be deducted if the final answers have inappropriate number of significant digits, no units or wrong units.

(8) The necessary values of fundamental and astronomical constants should be taken from the Table of Constants provided to you.

(9) Instructions regarding using the answersheets

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

b. You should also show detailed solution in the blank space provided on each Summary Answersheet. If necessary, you can ask for extra blank sheets from the invigilators.

c. Use separate blank sheets for each question.

d. Write your contestant code, question number and page number on FRONT side of each blank sheet. Write your contestant code and page number on the BACK side of each blank sheet.

e. The page numbers should be continuous, i.e., if you use 20 sheets for the entire examination (including the Summary Answersheets), the page numbers should run from 1 to 40.

f. Write only inside the boxed area.

g. For the work that you do not want to be evaluated, cross that part out.

(10) At the end of the examination

a. On the Cover Sheet, clearly write page numbers for each question.

b. Check that you have written your contestant code on all pages.

c. Put the Cover Sheet, all Summary Answersheets, blank sheets and rough sheets inside the envelope. You may keep question paper and Table of Constants with you.

Table of Constants

Fundamental Constants

Name	Symbol	Value
Speed of light in vacuum	$c$	$2.998\times10^8\ \mathrm{m\,s^{-1}}$
Planck constant	$h$	$6.626\times10^{-34}\ \mathrm{J\,s}$
Boltzmann constant	$k_B$	$1.381\times10^{-23}\ \mathrm{J\,K^{-1}}$
Stefan-Boltzmann constant	$\sigma$	$5.670\times10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}}$
Elementary charge	$e$	$1.602\times10^{-19}\ \mathrm{C}$
Gravitational constant	$G$	$6.674\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}$
Gas constant	$R$	$8.315\ \mathrm{J\,mol^{-1}\,K^{-1}}$
Avogadro constant	$N_A$	$6.022\times10^{23}\ \mathrm{mol^{-1}}$
Wien's displacement constant	$\lambda_mT$	$2.898\times10^{-3}\ \mathrm{m\,K}$
Mass of electron	$m_e$	$9.109\times10^{-31}\ \mathrm{kg}$
Mass of proton	$m_p$	$1.673\times10^{-27}\ \mathrm{kg}$
Mass of neutron	$m_n$	$1.675\times10^{-27}\ \mathrm{kg}$
Atomic mass unit	$u$	$1.661\times10^{-27}\ \mathrm{kg}$

Astronomical Constants

Name	Symbol	Value
Parsec	pc	$3.086\times10^{16}\ \mathrm{m}$
Astronomical unit	$a_\oplus$	$1.496\times10^{11}\ \mathrm{m}$
Mass of the Sun	$M_\odot$	$1.989\times10^{30}\ \mathrm{kg}$
Radius of the Sun	$R_\odot$	$6.955\times10^8\ \mathrm{m}$
Luminosity of the Sun	$L_\odot$	$3.826\times10^{26}\ \mathrm{W}$
Apparent magnitude of the Sun at midday		$-26.72\ \mathrm{mag}$
Solar constant		$1366\ \mathrm{W\,m^{-2}}$
Angular diameter of the Sun		$30'$
Mass of the Earth	$M_\oplus$	$5.972\times10^{24}\ \mathrm{kg}$
Radius of the Earth	$R_\oplus$	$6.371\times10^6\ \mathrm{m}$
Tropical year		$365.242$ solar days $=3.156\times10^7\ \mathrm{s}$

Условие

India astronomy satellite, AstroSat, launched in September 2015, has five different instruments.

AstroSat with UVIT, LAXPC, CZTI, SSM and SXT instruments labelled.

T12.1

In this question, we will discuss three of these instruments (SXT, LAXPC, CZTI), which point in the same direction and observe in X-ray wavelengths. The details of these instruments are given in the table below.

Instrument	Band [keV]	Collecting Area [m $^2$ ]	Effective Photon Detection Efficiency	Saturation level [counts]	No. of Pixels
SXT	0.3 – 80	0.067	60%	15000 (total)	512 x 512
LAXPC	3 – 80	1.5	40%	50000 (in any one counter) or 200000 (total)	---
CZTI	10 – 150	0.09	50%	---	4 x 4096

You should note that LAXPC energy range is divided into 8 different energy band counters of equal bandwidth with no overlap.

T12.1 Some X-ray sources like Cas A have a prominent emission line at 0.01825 nm corresponding to a radioactive transition of $^{44}$ Ti. Suppose there exists a source which emits only one bright emission line corresponding to this transition. What should be the minimum relative velocity ( $\nu$ ) of the source, which will make the observed peak of this line to get registered in a different energy band counter of LAXPC as compared to a source at rest? [13 т.]

T12.2–T12.6

These instruments were used to observe an X-ray source (assumed to be a point source), whose energy spectrum followed the power law,

F(E) = KE^{-2/3} \qquad \left[\text{in units of counts/keV/m}^2/\text{s}\right]

where $E$ is the energy in keV, $K$ is a constant and $F(E)$ is photon flux density at that energy. Photon flux density, by definition, is given for per unit collecting area (m $^2$ ) per unit bandwidth (keV) and per unit time (seconds). From prior observations, we know that the source has a flux density of 10 counts/keV/m $^2$ /s at 1 keV, when measured by a detector with 100% photon detection efficiency. The "counts" here mean the number of photons reported by the detector.

As the source flux follows the power law given above, we know that for a given energy range from $E_1$ (lower energy) to $E_2$ (higher energy) the total photon flux ( $F_{\mathrm{T}}$ ) will be given by

F_{\mathrm{T}} = 3K\left(E_2^{1/3} - E_1^{1/3}\right) \qquad \left[\text{in units of counts/m}^2/\text{s}\right]

T12.2 Estimate the incident flux density from the source at 1 keV, 5 keV, 40 keV and 100 keV. Also estimate what will be the total count per unit bandwidth recorded by each of the instruments at these energies for an exposure time of 200 seconds. [8 т.]

T12.3 For this source, calculate the maximum exposure time ( $t_{\mathrm{S}}$ ), without suffering from saturation, for the CCD of SXT. [4 т.]

T12.4 If the source became 3500 times brighter, calculate the expected counts per second in LAXPC counter 1, counter 8 as well as total counts across the entire energy range. If we observe for longer period, will the counter saturate due to any individual counter or due to the total count? Tick the appropriate box in the Summary Answersheet. [8 т.]

T12.5 Assume that the counts reported by CZTI due to random fluctuations in electronics are about 0.00014 counts per pixel per keV per second at all energy levels. Any source is considered as "detected" when the SNR (signal to noise ratio) is at least 3. What is minimum exposure time, $t_c$ , needed for the source above to be detected in CZTI?
Note that the "noise" in a detector is equal to the square root of the counts due to random fluctuations. [10 т.]

T12.6 Let us consider the situation where the source shows variability in number flux, so that the factor $K$ increases by 20%. AstroSat observed this source for 1 second before the change and 1 second after this change in brightness. Calculate the counts measured by SXT, LAXPC and CZTI in both the observations. Which instrument is best suited to detect this change? Tick the appropriate box in the Summary Answersheet. [7 т.]

Решение

Внимание: Непълно решение

Оригинал в Архива: theory-qp.pdf · официални решения: theory_sol.pdf · допълнителни данни: tableofconstants.pdf · допълнителни данни: tableofconstants.pdf · допълнителни данни: Instructions_theory.pdf · допълнителни данни: Instructions_theory.pdf · допълнителни данни: theory.pdf · допълнителни данни: theory.pdf