| exam,exam_stage,subject,domain,subdomain,context,image_context,image,questions,answer | |
| dc,standalone,maths,algebra,functions,"If $g(x) = k \cdot f(x)$ , what is the value of $k$ ?",,images/17_0.jpg,"[""If $g(x) = k \\cdot f(x)$ , what is the value of $k$ ?""]",-3 | |
| olympiad,final,astrophysics,stellar_astrophysics,binary_stars,"Astronomers use an 8-meter telescope to observe a star with an apparent flux of $3.068 \times 10^{- 9} \mathrm{W / m^2}$. A spectral analysis reveals a blackbody spectrum with two apparent peaks, one at $690.5 \mathrm{nm}$ and the other at $461.8 \mathrm{nm}$. Sustained spectroscopic observations of the hydrogen alpha line (rest wavelength $656.3 \mathrm{nm}$) results in the following plot, in which two periodic variations have been identified and marked. Note: the x-axis is in days, and the y-axis is in nanometers. | |
| Precision photometry of the system shows what appears to be an eclipsing binary light curve. The primary transit lasts 4 hours 46 minutes total. Maximum transit depth lasts for 4 hours 7 minutes.",,images/3_0.jpg,"[""(a) Calculate the ratio of the radii and the ratio of the luminosities of the two stars in the system. [4]""]","Using the transit light curve, the ratio of the radii is $\frac{r_1}{r_2} = 13.67$. | |
| Using Wien's Law and the Stefan-Boltzmann Law, the ratio of the luminosities is $\frac{L_1}{L_2} = 934$." | |
| olympiad,final,astrophysics,celestial_mechanics,keplers_laws,"Astronomers use an 8-meter telescope to observe a star with an apparent flux of $3.068 \times 10^{- 9} \mathrm{W / m^2}$. A spectral analysis reveals a blackbody spectrum with two apparent peaks, one at $690.5 \mathrm{nm}$ and the other at $461.8 \mathrm{nm}$. Sustained spectroscopic observations of the hydrogen alpha line (rest wavelength $656.3 \mathrm{nm}$) results in the following plot, in which two periodic variations have been identified and marked. Note: the x-axis is in days, and the y-axis is in nanometers. | |
| Precision photometry of the system shows what appears to be an eclipsing binary light curve. The primary transit lasts 4 hours 46 minutes total. Maximum transit depth lasts for 4 hours 7 minutes.",,images/3_0.jpg,"[""(b) Calculate the semimajor axis and mass of each star. [5]""]","Using the provided chart, the orbital period is approximately 5.7 days. | |
| Using the velocity of each star, the semimajor axes are 0.0450 and 0.0771 AU for stars 1 and 2 respectively. | |
| Applying Kepler's Third Law, the total mass is 1.90 solar masses. | |
| The individual masses are 1.20 solar masses for star 1 and 0.70 solar masses for star 2." | |
| olympiad,final,astrophysics,stellar_astrophysics,stellar_structure,"Astronomers use an 8-meter telescope to observe a star with an apparent flux of $3.068 \times 10^{- 9} \mathrm{W / m^2}$. A spectral analysis reveals a blackbody spectrum with two apparent peaks, one at $690.5 \mathrm{nm}$ and the other at $461.8 \mathrm{nm}$. Sustained spectroscopic observations of the hydrogen alpha line (rest wavelength $656.3 \mathrm{nm}$) results in the following plot, in which two periodic variations have been identified and marked. Note: the x-axis is in days, and the y-axis is in nanometers. | |
| Precision photometry of the system shows what appears to be an eclipsing binary light curve. The primary transit lasts 4 hours 46 minutes total. Maximum transit depth lasts for 4 hours 7 minutes.",,images/3_0.jpg,"[""(c) Calculate the actual luminosity and radius of each star. [3]""]","Using the velocity of each star, the radii are 1.35 and 0.099 solar radii for stars 1 and 2 respectively. | |
| Using the Stefan-Boltzmann Law, the luminosities are 2.54 and 0.0027 solar luminosities for stars 1 and 2 respectively." | |
| olympiad,final,astrophysics,stellar_astrophysics,stellar_structure,"Astronomers use an 8-meter telescope to observe a star with an apparent flux of $3.068 \times 10^{- 9} \mathrm{W / m^2}$. A spectral analysis reveals a blackbody spectrum with two apparent peaks, one at $690.5 \mathrm{nm}$ and the other at $461.8 \mathrm{nm}$. Sustained spectroscopic observations of the hydrogen alpha line (rest wavelength $656.3 \mathrm{nm}$) results in the following plot, in which two periodic variations have been identified and marked. Note: the x-axis is in days, and the y-axis is in nanometers. | |
| Precision photometry of the system shows what appears to be an eclipsing binary light curve. The primary transit lasts 4 hours 46 minutes total. Maximum transit depth lasts for 4 hours 7 minutes.",,images/3_0.jpg,"[""(d) How far away is the system? The Solar flux at Earth is 1366 W/m². [4]""]","Using the apparent magnitude and the solar flux, the distance is 5.94 pc." | |
| olympiad,final,astrophysics,observational_astronomy,telescopes_instrumentation,"Astronomers use an 8-meter telescope to observe a star with an apparent flux of $3.068 \times 10^{- 9} \mathrm{W / m^2}$. A spectral analysis reveals a blackbody spectrum with two apparent peaks, one at $690.5 \mathrm{nm}$ and the other at $461.8 \mathrm{nm}$. Sustained spectroscopic observations of the hydrogen alpha line (rest wavelength $656.3 \mathrm{nm}$) results in the following plot, in which two periodic variations have been identified and marked. Note: the x-axis is in days, and the y-axis is in nanometers. | |
| Precision photometry of the system shows what appears to be an eclipsing binary light curve. The primary transit lasts 4 hours 46 minutes total. Maximum transit depth lasts for 4 hours 7 minutes.",,images/3_0.jpg,"[""(e) What is the angular size of the stars' orbit? Can the telescope distinguish the two stars? What is the minimum separation the telescope can distinguish at the distance of this system? Assume the telescope observes at a wavelength of 550 nm. [3]""]","The maximum angular separation is 0.021 degrees. | |
| The best possible resolution of the telescope is 0.017 degrees, so the stars are distinguishable. | |
| The smallest visible size is 0.10 AU." | |
| olympiad,final,astrophysics,high_energy_astrophysics,accretion_physics,"Astronomers use an 8-meter telescope to observe a star with an apparent flux of $3.068 \times 10^{- 9} \mathrm{W / m^2}$. A spectral analysis reveals a blackbody spectrum with two apparent peaks, one at $690.5 \mathrm{nm}$ and the other at $461.8 \mathrm{nm}$. Sustained spectroscopic observations of the hydrogen alpha line (rest wavelength $656.3 \mathrm{nm}$) results in the following plot, in which two periodic variations have been identified and marked. Note: the x-axis is in days, and the y-axis is in nanometers. | |
| Precision photometry of the system shows what appears to be an eclipsing binary light curve. The primary transit lasts 4 hours 46 minutes total. Maximum transit depth lasts for 4 hours 7 minutes.",,images/3_0.jpg,"[""(f) Does the available data indicate any other objects in the system? If so, provide mass, semimajor axis, and a likely type. Justify your identification. [6]""]","Examining the given plot, there appears to be a longer period variation with a period of approximately 214 days and an amplitude of 83.2 km/s. | |
| Using this velocity and the period of oscillation, the semimajor axis is 1.62 AU. | |
| The unknown object has a mass of 16.2 solar masses and a semimajor axis of 0.19 AU. | |
| Given the high mass and lack of a visible counterpart, the object is likely a stellar mass black hole." | |