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IOAA 2016, Теория, английски — Задача 11. Gravitational Waves

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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

The first signal of gravitational waves was observed by two advanced LIGO detectors at Hanford and Livingston, USA in September 2015. One of these measurements (strain vs time in seconds) is shown in the accompanying figure. In this problem, we will interpret this signal in terms of a small test mass $m$ orbiting around a large mass $M$ (i.e., $m \ll M$ ), by considering several models for the nature of the central mass.

GW150914 strain versus time from LIGO Louisiana; all axes, merger marker and source attribution.

The test mass loses energy due to the emission of gravitational waves. As a result the orbit keeps on shrinking, until the test mass reaches the surface of the object, or in the case of a black hole, the innermost stable circular orbit – ISCO – which is given by $R_{\mathrm{ISCO}} = 3R_{\mathrm{sch}}$ , where $R_{\mathrm{sch}}$ is the Schwarzschild radius of the black hole. This is the "epoch of merger". At this point, the amplitude of the gravitational wave is maximum, and so is its frequency, which is always twice the orbital frequency. In this problem, we will only focus on the gravitational waves before the merger, when Kepler's laws are assumed to be valid. After the merger, the form of gravitational waves will drastically change.

T11.1 Consider the observed gravitational waves shown in the figure above. Estimate the time period, $T_0$ , and hence calculate the frequency, $f_0$ , of gravitational waves just before the epoch of merger. [3 т.]

T11.2 For any main sequence (MS) star, the radius of the star, $R_{\mathrm{MS}}$ , and its mass, $M_{\mathrm{MS}}$ , are related by a power law given as,

R_{\mathrm{MS}} \propto (M_{\mathrm{MS}})^{\alpha}

where $\alpha = 0.8$ for $M_{\odot} < M_{\mathrm{MS}}$ and $\alpha = 1.0$ for $0.08M_{\odot} \leq M_{\mathrm{MS}} \leq M_{\odot}$ .

If the central object were a main sequence star, write an expression for the maximum frequency of gravitational waves, $f_{\mathrm{MS}}$ , in terms of mass of the star in units of solar masses ( $M_{\mathrm{MS}}/M_{\odot}$ ) and $\alpha$ . [10 т.]

T11.3 Using the above result, determine the appropriate value of $\alpha$ that will give the maximum possible frequency of gravitational waves, $f_{\mathrm{MS,max}}$ for any main sequence star. Evaluate this frequency. [9 т.]

T11.4 White dwarf (WD) stars have a maximum mass of $1.44\,M_{\odot}$ (known as the Chandrasekhar limit) and obey the mass-radius relation $R \propto M^{-1/3}$ . The radius of a solar mass white dwarf is equal to 6000 km. Find the highest frequency of emitted gravitational waves, $f_{\mathrm{WD,max}}$ , if the test mass is orbiting a white dwarf. [8 т.]

T11.5 Neutron stars (NS) are a peculiar type of compact objects which have masses between 1 and $3M_{\odot}$ and radii in the range 10 – 15 km. Find the range of frequencies of emitted gravitational waves, $f_{\mathrm{NS,min}}$ and $f_{\mathrm{NS,max}}$ , if the test mass is orbiting a neutron star at a distance close to the neutron star radius. [8 т.]

T11.6 If the test mass is orbiting a black hole (BH), write the expression for the frequency of emitted gravitational waves, $f_{\mathrm{BH}}$ , in terms of mass of the black hole, $M_{\mathrm{BH}}$ , and the solar mass $M_{\odot}$ . [7 т.]

T11.7 Based only on the time period (or frequency) of gravitational waves before the epoch of merger, determine whether the central object can be a main sequence star (MS), a white dwarf (WD), a neutron star (NS), or a black hole (BH). Tick the correct option in the Summary Answersheet. Estimate the mass of this object, $M_{\mathrm{obj}}$ , in units of $M_{\odot}$ . [5 т.]

Решение

T11.1

Solution:

From the graph, just before the peak of emission, the time period of gravitational waves is approximately $(0.007 \pm 0.004)\ \mathrm{s}$ .

T_0 \approx 0.007\ \mathrm{s}

[2.0]

Acceptable range: 0.003 to 0.011 s

That is, the frequency of the gravitational waves is $f_0 \approx 142.86\ \mathrm{Hz}$ . [1.0]

Acceptable range: 333.33 to 90.91 Hz

Answer given in terms of angular frequency with correct value and units gains full credit.

T11.2

Solution:

Since $m \ll M$ then, by Kepler's third law

f_{\mathrm{orbital}} = \frac{1}{2\pi}\sqrt{\frac{GM}{r^3}}

[4.0]

Hence the frequency of the gravitational waves is

f_{\mathrm{grav}} = 2f_{\mathrm{orbital}} = \frac{1}{\pi}\sqrt{\frac{GM}{r^3}}

[1.0]

The frequency will be maximum when $r = R_{\mathrm{MS}}$ . [1.0]

For main sequence stars,

\frac{R_{\mathrm{MS}}}{R_{\odot}} = \left(\frac{M_{\mathrm{MS}}}{M_{\odot}}\right)^{\alpha}

\therefore\ R_{\mathrm{MS}} = R_{\odot}\left(\frac{M_{\mathrm{MS}}}{M_{\odot}}\right)^{\alpha}

[1.0]

\therefore\ f_{\mathrm{MS}} = \frac{1}{\pi}\sqrt{\frac{GM_{\mathrm{MS}}}{R_{\odot}^3}}\left(\frac{M_{\odot}}{M_{\mathrm{MS}}}\right)^{3\alpha/2}

= \frac{1}{\pi}\sqrt{\frac{GM_{\odot}}{R_{\odot}^3}}\left(\frac{M_{\odot}}{M_{\mathrm{MS}}}\right)^{(3\alpha-1)/2}

f_{\mathrm{MS}} = \frac{1}{\pi}\sqrt{\frac{GM_{\odot}}{R_{\odot}^3}}\left(\frac{M_{\mathrm{MS}}}{M_{\odot}}\right)^{(1-3\alpha)/2}

[3.0]

T11.3

Solution:

For possible values of $\alpha$ given in the question, the exponent $\frac{1-3\alpha}{2}$ is negative. Thus, if $M_{\mathrm{MS}} > M_{\odot}$ , the frequency will be smaller. Thus, for highest possible frequency coming from a main sequence star, you should take lowest possible mass i.e. $\alpha = 1.0$ [4.0]

f_{\mathrm{MS,max}} = \frac{1}{\pi}\sqrt{\frac{GM_{\odot}}{R_{\odot}^3}}\left(\frac{M_{\mathrm{MS}}}{M_{\odot}}\right)^{\left(\frac{1-3\times 1}{2}\right)}

= \frac{1}{\pi}\sqrt{\frac{GM_{\odot}}{R_{\odot}^3}} \times \frac{M_{\odot}}{M_{\mathrm{MS}}}

[2.0]

The frequency of gravitational waves will be given by,

f_{\mathrm{MS,max}} = \frac{1}{\pi}\sqrt{\frac{6.674 \times 10^{-11} \times 1.989 \times 10^{30}}{(6.955 \times 10^8)^3}} \times \frac{1}{0.08}

f_{\mathrm{MS,max}} = 2.5\ \mathrm{mHz}

[3.0]

Answer given in terms of angular frequency with correct value and units gains full credit.

T11.4

Solution:

The maximum frequency would be when $r = R_{\mathrm{WD}}$ . [1.0]

We use the notation $R_{WD_{\odot}}$ for the radius of a solar mass white dwarf. Then for white dwarfs

R_{WD}^3 = R_{WD_{\odot}}^3\,\frac{M_{WD}}{M_{\odot}}

[1.0]

f_{\mathrm{WD}} = \frac{1}{\pi}\sqrt{\frac{GM_{\mathrm{WD}}}{R_{WD}^3}}

= \frac{1}{\pi}\sqrt{\frac{GM_{\odot}}{R_{WD_{\odot}}^3}}\,\frac{M_{WD}}{M_{\odot}}

[2.0]

For maximum frequency, we have to take highest white dwarf mass. [2.0]

f_{\mathrm{WD,max}} = 2.600 \times 10^{-6} \times \sqrt{\frac{1.989 \times 10^{30}}{(6000 \times 10^3)^3}} \times 1.44

= 2.600 \times 10^{-6} \times 95.96 \times 10^3 \times 1.44

f_{\mathrm{WD,max}} = 0.359\ \mathrm{Hz}

[2.0]

Answer given in terms of angular frequency with correct value and units gains full credit.

T11.5

Solution:

f_{grav} = \frac{1}{\pi}\sqrt{\frac{GM_{\mathrm{NS}}}{R_{\mathrm{NS}}^3}}

The lowest possible frequency is when $M_{\mathrm{NS}}$ is lowest and $R_{\mathrm{NS}}$ is the highest. [2.0]

For $M_{\mathrm{NS}} = M_{\odot}$ and $R = 15\ \mathrm{km}$ , we get

f_{\mathrm{NS,min}} = 1.996\ \mathrm{kHz}

[2.0]

Similarly, the largest possible frequency is when $M_{\mathrm{NS}}$ is the largest and $R_{\mathrm{NS}}$ is the smallest. [2.0]

For $M_{\mathrm{NS}} = 3M_{\odot}$ and $R = 10\ \mathrm{km}$ , we get

f_{\mathrm{NS,max}} = 6.352\ \mathrm{kHz}

[2.0]

Answer given in terms of angular frequency with correct value and units gains full credit.

T11.6

Solution:

For black holes, we have to consider $R_{ISCO}$ . [2.0]

Hence the equation will be,

f_{BH} = \frac{1}{\pi} \times \sqrt{\frac{GM_{\odot}}{27R_{sch-\odot}^3}} \times \frac{M_{\odot}}{M_{BH}}

[2.0]

f_{BH} = 4.396\ \mathrm{kHz} \times \frac{M_{\odot}}{M_{BH}}

[3.0]

Answer given in terms of angular frequency with correct value and units gains full credit.

T11.7

Solution:

We found the frequency of the LIGO-detected wave to be 166.67 Hz just before merger. As per our analysis above, only black holes can lead to emission in this frequency range.

Black Hole [2.0]

By using corresponding expression,

M_{\mathrm{obj}} = \frac{4396}{142.86}\,M_{\odot} \approx 31\,M_{\odot}

[3.0]

Any answer between 13 to 50 will get full credit.

The key restates the GW150914 strain graph, with axes and source attribution.

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