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USAAAO 2020, II кръг — Задача 10

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от USAAAO 2020

2020 National Astronomy Competition · 15 т.

Instructions (Please Read Carefully)

The top 5 eligible scorers on the NAC will be invited to represent USA at the next IOAA. The next 5 eligible scorers will be invited to join the USA guest team at the next IOAA. In order to qualify for the national team, you must be a high school student with US citizenship or permanent residency.

This exam consists of 3 parts: Short Questions, Medium Questions and Long Questions.

The test must be completed within 2.5 hours (150 minutes).

Please solve each problem on a blank piece of paper and mark the number of the problem at the top of the page. The contestant’s full name in capital letters should appear at the top of each solution page. If the contestant uses scratch papers, those should be labeled with the contestant’s name as well and marked as “scratch paper” at the top of the page. Scratch paper will not be graded. Partial credit will be available given that correct and legible work was displayed in the solution.

This is a written exam. Contestants can only use a scientific calculator for this exam. A table of physical constants will be provided. Discussing the problems with other people is strictly prohibited in any way until the end of the examination period on April 5th. Receiving any external help during the exam is strictly prohibited. This means that the only allowed items are: a scientific calculator, the provided table of constants, a pencil (or pen), an eraser, blank sheets of papers, and the exam. No books or notes are allowed during the exam.

Условие

The following table gives the numerical values for some physical properties of four stars. The quantities that are affected by, i.e. include the effects of, interstellar extinction are marked with a star (*). You may consider that all stars are black bodies. The temperature of a star can be calculated directly from its B-V index, by using Ballesteros’ formula:

T_{eff} = f(B-V) = 4600\left(\frac{1}{0.92(B-V) + 1.7} + \frac{1}{0.92(B-V) + 0.62}\right)K.

Determine the numerical values of all the other physical characteristics presented in the given table. For full credit, show your full work by writing all the mathematical expressions used in the calculation.

Hint: You might use the following empirical relation:

\frac{A_V}{E_{B-V}} = 3.2

Star	$\kappa$ Velorum	$\beta$ Tauri	Sirius A	Sun
Annual Parallax $p^*(10^{-3}arcsec)$	6.05	24.89	379.2	-
Distance to Sun $\Delta^*(pc)$				-
Interstellar Extinction in V Band $A_V$ (mag)	0.20	0.08	Negligible	-
$10^{0.2\times A_V}$				-
Distance to Sun $\Delta(pc)$				$4.85 \times 10^{-6}$
Annual Parallax $p(10^{-3}$ arcsec)
Distance Modulus $\mu = m - M$
Visual apparent magnitude $m^*(mag)$	2.86	1.68	-1.47	-26.73
Visual apparent magnitude m (mag)
Visual absolute magnitude $M_V$ (mag)
Color Index $(B-V)^*$ (mag)	-0.14	-0.06	+0.01	+0.65
Extinction $E_{B-V}$ (mag)
Color Index $(B-V)$ (mag)
Effective Temperature $T_{eff} = f(B-V)$ (K)
$\lambda_m$ (nm)
Radius (Solar Radius, $R_S$ )	9.10	4.60	1.71	1.00
Total Luminosity (Solar Luminosity, $L_S$				1.00
Absolute Bolometric Magnitude $M_{bol}$ (mag)				4.64
Bolometric Correction BC for V band (mag)				-0.20

Отговори

Решение

Solution:
It is known that:

\Delta^*_{pc} = \frac{1}{p^*_{arcsec}}

Also,

m = M + 5\log\Delta_{pc} - 5

without extinction, and

m^* = M + 5\log\Delta^*_{pc} - 5

with interstellar extinction. Because the luminous flux is dimmed by the interstellar extinction,

m^* = m + A_V

Thus,

\Delta_{pc} = \Delta^*_{pc} \times 10^{-0.2A_V}

Distance modulus is given by:

\mu = m - M = 5\log\Delta_{pc} - 5

And thus, the absolute visual magnitude can be computed as:

M_V = m - \mu

For the Color Excess, we know that:

E_{B-V} = (B-V)^* - (B-V)

In order to find the temperature of a given star, we can use Ballesteros’ formula, given in the question. Next, from Wien’s formula:

\lambda \times T_{eff} = b

Moreover,

L_s = \sigma \times T^4 \times 4\pi \times R^2

So, we can show that:

M_{BOL} - M_{BOL_{Sun}} = -2.5\log\frac{L}{L_{Sun}}

Lastly, the Bolometric Correction is:

B.C. = M_{BOL} - M_V

Star	$\kappa$ Velorum	$\beta$ Tauri	Sirius A	Sun
Annual Parallax $p^*(10^{-3}$ arcsec)	6.05	24.89	379.2	-
Distance to Sun $\Delta^*$ (pc)	165.28	40.17	2.63	-
Interstellar Extinction in V Band $A_V$ (mag)	0.20	0.08	Negligible	-
$10^{0.2\times A_V}$	1.10	1.04	1	-
Distance to Sun $\Delta(pc)$	150.26	38.63	2.63	$4.85 \times 10^{-6}$
Annual Parallax p ( $10^{-3}$ arcsec)	6.655	25.880	379.200	286.05
Distance Modulus $\mu = m - M$	5.89	2.93	-2.89	-31.57
Visual apparent magnitude $m^*$ (mag)	2.86	1.68	-1.47	-26.73
Visual apparent magnitude m (mag)	2.26	1.60	-1.47	-27.73
Visual absolute magnitude $M_V$ (mag)	-3.63	-1.33	1.42	4.84
Color Index $(B-V)^*$ (mag)	-0.14	-0.06	+0.01	+0.65
Extinction $E_{B-V}$ (mag)	0.0625	0.025	negligible	-
Color Index (B-V) (mag)	-0.2025	-0.085	+0.01	+0.65
Effective Temperature $T_{eff} = f(B-V)$ (K)	13616	11270	9936	5750
$\lambda_m$ (nm)	220.00	266.19	301.93	521.73
Radius (Solar Radius, $R_S$ )	9.100	4.600	1.711	1.000
Total Luminosity (Solar Luminosity, $L_S$	2350	277	23	1.00
Absolute Bolometric Magnitude $M_{bol}$ (mag)	-3.79	-1.48	+1.21	4.64
Bolometric Correction BC for V band (mag)	-0.16	-0.15	-0.20	-0.20

Оригинал в Архива: usaaao_second_exam_2020__apr4.pdf · официални решения: usaaao_second_exam_2020__apr4sol.pdf