USA
ASTRONOMY & ASTROPHYSICS
OLYMPIAD
2015
SECOND ROUND
National Astronomy Olympiad
2014-‐2015
This is the Second Round of the 2014-‐2015 USA Astronomy and Astrophysics Olympiad (USAAAO) competitions. You have exactly 2 hours and 30 minutes to complete the exam. There is a maximum of 67 points, and the value of each problem is indicated following each problem statement.
This test consists of 13 short answer problems and one long problem with multiple parts. Solutions should be written on blank paper, with the problem number,
student’s name, and school written clearly on top of each page. Solutions to multiple short answer problems may be written on the same page, but the long problem should begin on its own page. Partial credit will be given for correct work, so make sure to clearly show all steps.
You may only use a scientific calculator as aid for this exam. A table of physical constants and other information will be provided for you. This exam document, your solutions, and all used scratch paper must be turned in at the end of the exam.
Do not discuss this examination with anyone after its completion. Your results will be emailed to you shortly. The top 5 students will be invited to represent the United States at the 9th International Olympiad on Astronomy and Astrophysics
competition in Central Java, Indonesia, from July 26th to August 4th, 2015.
In the event of a tie for the top 5 places, the student with the higher score on the long problem will qualify. If the tied students have the same score on the long problem, then the short answer problems with more points will be given more weight in the grading.
I hereby affirm that all work on this exam is mine, and that I took this test under a proctor’s supervision, with no outside aids beyond the materials provided and allowed. Furthermore, I affirm to not discuss the test with others or provide any sort of aid to other examiners throughout the course of the examination period. I
understand that failure to do so may result in disqualification from the exam.
Signature: ______________________________________________ Date: ____________________
Section A: Short Answer [42 Points]
1. A blue star with effective temperature 𝑇eff= 10,000 K and apparent
magnitude 𝑚 = 5 is located 150 pc from Earth. Find the radius of the star. [2]
2. The radial velocity curve of a nearby Solar-‐mass star shows that it has a planet orbiting it with a period of 3 days. This planet causes its host star to have a radial velocity semi-‐amplitude of 50 m/s. Assuming that this planet’s orbit is perfectly inclined to Earth, and has 0 eccentricity, what is the mass of this planet, in Jupiter masses? [2]
3. The Sun has a rotation rate of 24.5 days. Jupiter has a mass of 9.54 × 10!!
solar masses, and a semimajor axis of 5.2 AU. Which object has more total angular momentum? [2]
4. What is the distance to a star cluster whose stars at the main sequence turn-‐ off point have an apparent magnitude of 𝑚 = 10 and an effective
temperature of 𝑇eff ≈ 6000 K? How old is this cluster? [3]
5. Assuming that the universe currently is well described by a density
parameter Ω0 = 1, there is no dark energy and the current temperature of the
universe is 2.73 K, compute how long from the present it will take for the universe to cool down by 0.2 K. Remember that the temperature of the universe is inversely proportional to its radius (the scale factor). [3]
6. A star has apparent magnitude 𝑚 = 8, parallax 𝑝 = 0.003" and effective temperature 𝑇eff ≈ 6000 K. What is the luminosity of the star? What is the
likely spectral type of this star? Justify your answer. [3]
7. Assuming that the cosmic microwave background radiation has the spectrum of a blackbody throughout the evolution of the universe, determine how its temperature changes with redshift. In particular, find the temperature of the CMB at the epoch z ≈ 10, knowing that the current temperature of the CMB is 2.73 K. [3]
8. Derive an expression for the blackbody temperature of a planet with radius 𝑟 and albedo 𝑎, orbiting its sun at a certain distance 𝐷. The star has a
temperature 𝑇∗ and radius 𝑅. You must show your derivation for credit, not just the final expression. [3]
9. A 2048 x 3072 pixel CCD camera with 7.2 micron pixels is attached to an f/10 telescope with a 0.256 m primary mirror. What is the angular resolution of the CCD, in arcseconds/pixel? [3]
10. An interplanetary spacecraft bound for Saturn (a = 9.6 AU) is launched into a 300 km, non-‐inclined parking orbit around the Earth. If the spacecraft takes a Hohmann transfer to Saturn, what is the delta-‐V required for trans-‐Saturn injection, and what is the required delta-‐V for insertion into a 100000 km circular orbit around Saturn? On what side of each planet should the burns occur? (Assume Saturn is in the same orbital plane as the Earth, and neglect the gravitational deflection of the ship’s path after injection and before orbital insertion.) [4]
11. Planet A orbits around Star A of mass 𝑀 = 0.54 𝑀⊙ with a period of 𝑃 = 6 Earth years. Astronomers on this planet want to measure the distance of a distant Star B which happens to lie along the semimajor axis of Planet A’s orbit, on the side of perihelion. They choose to do so using parallax, by noting the position of Star B with respect to the background stars at two different points in the orbit. These two points, X and Y, are located such that XY is perpendicular to the semimajor axis, and intersects it at the focus, i.e. where Star A is located. Astronomers measure the angle that Star B appears to change from X to Y as 𝜃 = 0.05".’ If Planet A is 0.537 AU from Star A at perihelion, what is the distance to Star B in parsecs? [4]
12. On the vernal equinox, the Sun has a right ascension of 0 hours and a
declination of 0°. Starting from the equinox, calculate the number of days, to the nearest tenth, that it takes for the Sun to have a right ascension of 4 hours. Assume a perfectly circular orbit and that the Earth is inclined 23.5° to the ecliptic plane. [4]
13. Rigel has equatorial coordinates RA = 05h 14m 32.3s, Dec = -‐08° 12’ 06”, and Betelgeuse has coordinates RA = 05h 55m 10.3s, Dec = +07° 24’ 25”. What is the angular separation between the two stars, and the position angle of Rigel relative to Betelgeuse? If a photographer wants to take a photo containing both stars, what is the maximum focal length they can use? [6]
Section B: Long Question [25 Points]
Astronomers use an 8-‐meter telescope to observe a star with an apparent flux of 3.068 × 10!! W/m!. A spectral analysis reveals a blackbody spectrum with two
apparent peaks, one at 690.5 nm and the other at 461.8 nm. Sustained spectroscopic observations of the hydrogen alpha line (rest wavelength 656.3 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.
(a) Calculate the ratio of the radii and the ratio of the luminosities of the two stars in the system. [4]
(b) Calculate the semimajor axis and mass of each star. [5]
(d) How far away is the system? The Solar flux at Earth is 1366 W/m!. [4]
(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]
(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]
USAAAO 2015 Second Round Solutions
Problem 1
T = 10000 K, m = 5, d = 150 pc m − M = 5 log(d/10)
M = m − 5 log(d/10) = 5 − 5 log(15) = −0.88
We compare this with the absolute magnitude of the sun, 4.83.
L
Lsolar = 100
(4.83−−0.88)/5= 192L solar
From the Stefan-Boltzmann Law, L = 4πR2σT4
Finding the ratio with the sun, we get:
L Lsolar = R Rsolar 2 T Tsolar 4 R Rsolar = L Lsolar 1/2 T solar T 2 = 4.6Rsolar Problem 2 T = 3 days, K = 50 m/s, M = 1Msolar T2=a3 M 3 365 2/3
= 0.041AU for the planet’s orbit. For the star’s orbit:
T = 2πr
v
r = T v2π = 1.38 ∗ 10−5AU
Relating the two, we have:
m∗r∗= mprp, so we have mp= rr∗
pm∗
Plugging in, we find mp= 3.36 ∗ 10−4Msolar
Problem 3
Solar rotation rate 24.5 days, MJ = 9.54 ∗ 10−4Msolar, a = 5.2 AU, solar
radius 695,000 km. L = Iω Lsolar =25M R2∗24.52π = 5 ∗ 1010 For Jupiter, L = mrv r = 5.2 ∗ 149.6 ∗ 106= 7.77 ∗ 108km v = 5.22/32πr∗365 = 1.13 ∗ 10 6km/day. L = 9.54 ∗ 10−4∗ 7.77 ∗ 108∗ 1.13 ∗ 06= 8.38 ∗ 1011
So Jupiter has the greater angular momentum.
Problem 4
m = 10, T = 6000 K at the main sequence turnoff.
Oldest main sequence stars are 6000 K, which is approximately sun-like. We therefore assume the absolute magnitude of the stars at the turnoff point is 4.83.
m − M = 5 log(d/10)
d = 10 ∗ 10(m−M )/5= 108pc
The stars at the turnoff point are sunlike, so we expect them to have a lifetime of 10 Gyr. Since these are the oldest main sequence stars in the cluster, the cluster has an age of approximately 10 Gyr.
Problem 5
Question not graded
Problem 6 m = 8, p = 0.003”, and T = 6000 K. d = 1/p = 333pc m − M = 5 log(d/10) M = m − 5 log(d/10) = 0.385 L Lsolar = 100 (Msolar−M )/5= 59.9L solar
The star’s temperature is approximately sunlike, suggesting class G, but it is significantly more luminous, suggesting a giant. G0III would be a reasonable possible spectral type.
Problem 7
From Wein’s Law, λ = Tb. From the definition of Redshift, z = λ−λ0
λ0 , where
λ0 is the emitted wavelength. Solving for λ0, we get λ0= λ/(z + 1)
Combining this with Wein’s Law, we get: T = b∗(z+1)lambda
Using Wein’s Law and the given temperature of 2.73 K, we find that the recieved wavelength is 1.06 mm. Plugging this into the above expression, we obtain at temperature of 30.03 K for z = 10.
Problem 8
At blackbody equilibrium, power in is equal to power out, so we have:
Pout= 4πR2pσT 4 p Pin= A ∗ (1 − α) ∗4πDL2 = πR2p(1 − α) 4πR2 ∗σT∗4 4πD2 Pin= Pout 4πR2pσTp4= πR2p(1 − α) 4πR2 ∗σT∗4 4πD2 Tp4= (1 − α)T∗4 R2 ∗ 4D2 Tp= (1 − α)1/4T∗ q R∗ 2D Problem 9 7.2 micron pixels, f/10, D = 0.256 m. f/10, so focal length is 2.56 m.
Angular resolution is given by f ocallengthpixelsize ∗ 206265
Plugging in, we get a resolution of 0.58 arcseconds/pixel
Problem 10
A Hohmann transfer orbit is being used to go from the 1 AU orbit of the Earth to Saturn’s orbit at 9.6 AU. The semimajor axis of the transfer orbit is thus 5.3 AU.
Using the vis-viva equation, v2
2 − µ r = −
µ
2a, this corresponds to a velocity of
When it starts, however, the spacecraft is already in Earth parking orbit, so benefits from both its orbital velocity around the Earth and the Earth’s orbital velocity around the Sun.
When in the parking orbit, the probe has a maximum velocity relative to
the Sun ofqGMe
rorb +
q
GMs
rearth = 37540m/s
The difference between the spacecraft’s current maximum velocity relative to the Sun and the orbital velocity of the transfer ellipse is equal to the required ∆v, since we’re neglecting further attraction from the Earth. The necessary ∆v is thus 2550 m/s, and the burn must occur on the night side of the Earth in order to increase the orbital radius and match Saturn’s orbit (since the parking orbit is prograde).
We use the vis-viva equation again to calculate the probe’s velocity at Sat-urn’s orbital distance, and compute SatSat-urn’s orbital velocity using the same
technique we used for Earth. We then subtract Saturn’s velocity from the
probe’s velocity to find the probe’s velocity relative to Saturn.
We then calculate the velocity relative to Saturn for a 100,000 km circular orbit. The difference between this value and the value above is the ∆v required to make Saturn orbit, coming out to 24900 m/s. The burn must occur on the night side of Saturn in order to enter a prograde orbit.
Problem 11
M = 0.54Ms, P = 6 years, perihelion distance 0.537 AU, parallax 0.05”.
From the parallax equation, d = r/θ, where d is the distance, r is the baseline, and θ is the parallax angle. We therefore need to find the baseline, XY, which is the latus rectum of the ellipse.
Using Kepler’s Third Law, we find that the semimajor axis is 2.69 AU. This corresponds to an aphelion of 2.135 AU, and therefore an eccentricity of e = 0.3.
The latus rectum of an ellipse is given by l = a(1 − e2). Plugging in, we get
l = 2.45 AU, yielding a distance d = 49.0 pc
Problem 12
Simply dividing the length of the year by 6 is incorrect. Because of the inclination of the Earth, the Sun also varies in declination (from -23.5 to +23.5 degrees), but maintains (for this problem) a constant angular velocity. Spherical trigonometry is therefore required to determine the actual angle that the Sun traverses going from 0 to 4 hours (62.1 degrees). Since the Sun has constant angular velocity, traversing 360 degrees per year, the number of days can be expressed as (62.1/360)*365 = 63.0 days.
Problem 13
This question also requires spherical trigonometry. Picking RA = 0, Dec = 90 is likely the easiest third point. Now that we have a spherical triangle, we can use the spherical law of sines or cosines to find the separation angle (18.59 degrees).
The position angle is measured east of north. For our chosen triangle, this corresponds to the angle with Betelgeuse as its vertex. Again applying the spherical trigonometric relationships, we find the position angle to be 33.12 degrees.
To cover both stars, the picture must cover 18.6 degrees of the sky, which corresponds to 66960 arcseconds. Plate scale, in arcseconds/mm, is given by 206265/focal length. Solving for the focal length, we get a value of 3.08*film size, which is a focal length of 108 mm on 35 mm film or 216 mm on 70 mm.
Long Problem
a. Drawing the transit light curve, we can see that between the first and second contacts, the relative motion of the stars is equal to twice the radius of the transiting star. Similarly, the time between second and third contacts is
proportional to twice the primary star’s radius. Dividing, we get r1
r2 =
t2−t1
t3−t2 =
13.67
We can use Wein’s Law and the provided blackbody peaks to calculate the temperature of each star. We can now use the Stefan-Boltzmann Law to find the ratio of the luminosities:
L1 L2 = R 1 R2 2T 1 T2 4 = 934
b. We use the provided chart to determine the orbital period of the binary
system (approximately 5.7 days) as well as the semi-amplitude (using ∆λλ =vc).
Using the period and velocity of each star, we determine the semimajor axes
according to r = vT2π, for semimajor axes of 0.0450 and 0.0771 AU for stars 1
and 2 respectively.
Applying Kepler’s Third Law for the entire system, we get a total mass of
1.90 solar masses. Since m1r1= m2r2, we can determine the individual mass of
each star, 1.20 solar masses for star 1 and 0.70 solar masses for star 2.
c. Using the velocity of each star, we can find the relative velocity of the two stars (just sum). This can then be used to solve the equations from part a directly, giving radii of 1.35 and 0.099 solar radii for stars 1 and 2, respectively. We can now apply the Stefan-Boltzmann Law, dividing by the solar expression, to determine the luminosity of each star (2.54 solar luminosities for star 1, 0.0027 for star 2).
d. Apparent magnitude can be found by finding the total flux of the system and using the Sun’s apparent magnitude and the solar flux as a standard
can-dle, applying ∆M = 2.512 logF1
F2. Similarly, using the luminosity of the system
with the Sun’s absolute magnitude as a standard candle can provide an abso-lute magnitude for the system. Now that we have an apparent and absoabso-lute magnitude, we apply the distance modulus to get d = 5.94 pc.
e. Applying the small angle formula, we get a maximum angular separation
of 0.021”. The best possible resolution of the telescope is given by θ = 1.22Dλ ∗
206265 = 0.017”, so the stars are distinguishable. The smallest visible size can be found by applying the small angle formula with the limiting resolution, and is 0.10 AU.
with a period of approximately 214 days. Based on the change in wavelength, the variation has an amplitude of approximately 83.2 km/s. Using this velocity and the period of oscillation, we can determine the semimajor axis of the binary stars’ orbit (1.62 AU)
We now apply the fact that m1a1 = m2a2 and Kepler’s third law for the
binary-unknown system, solving for a2 and m2. We find that 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.
USAAAO First Round 2015
This round consists of 30 multiple-choice problems to be completed in 75 minutes. You may only use a scientific calculator and a table of constants during the test. The top 50% will qualify for the Second Round.
1. At arms length, the width of a fist typically subtends how many degrees of arc? a. 1o b. 5o c. 10o d. 15o e. 20o 2. To have a lunar eclipse, the line of nodes must be pointing at the sun. The moon must also be in what phase? a. New b. First Quarter c. Waxing Gibbous d. Full e. Waning Crescent 3. Mars orbits the sun once every 687 days. Suppose Mars is currently in the constellation Virgo. What constellation will it most likely be in a year from now? a. Virgo b. Scorpius c. Aquarius d. Taurus e. Cancer 4. To calculate the field of view of a telescope, you measure the time it takes Capella
(RA:5.27h, dec:45.98o) to pass across the eyepiece. If the measured time is 2 minutes and
30 seconds, what is the field of view in arcseconds? a. 11.6’ b. 26.5’ c. 37.5’ d. 52.5’ e. 66.8 5. A telescope with focal length of 20 mm and aperture of 10 mm is connected to your smartphone, which has a CCD that measures 4.0mm by 4.0mm. The CCD is 1024 by 1024 pixels. Which is closest to the field of view of the telescope?
b. 5o c. 10o d. 15o e. 20o 6. What is its the resolution in arcseconds per pixel? a. 10”/pixel b. 40”/pixel c. 120”/pixel d. 1200”/pixel e. 3600’/pixel 7. Comet 67P/Churyumov–Gerasimenko has an orbital period around the Sun of 6.44 years. What is its semimajor axis, in AU? a. 41.47 b. 16.34 c. 6.44 d. 3.46 e. 1.86 8. Which of the following techniques most directly constrains the mass of an exoplanet? a. Radial Velocity b. Transit Timing c. Microlensing d. Direct Imaging e. Proper Motion 9. Which two properties of galaxies does the TullyFisher relation utilize a correlation between? a. Luminosity and velocity dispersion b. Luminosity and rotational velocity c. Radius and metallicity d. Luminosity and metallicity e. Mass and surface brightness 10. A binary star system has two components: Star A and Star B. Star A has a mass of 5 solar masses, and Star B has the same mass as our Sun. Assuming circular orbits, how many times closer to the center of mass of the system is Star A than Star B? a. 1 b. 3 c. 5 d. 10 e. 25
11. What is, approximately, the peak wavelength of electromagnetic radiation emitted by a star at a temperature of 5,000 K? a. 580 Angstroms b. 5,800 Angstroms c. 4,600 Angstroms d. 2,900 Angstroms e. 58,000 Angstroms 12. Stars A and B are observed over a period of 1 year. Both stars appear to move with respect to the background stars from the position indicated on the left in the diagram below, to the position indicated on the right, and then back to the position on the left over the full year. Which star is further from the Earth? a. Star A b. Star B c. Both stars are the same distance from the Earth d. Not enough information given 13. Suppose that you measure the parallax angle for a particular star to be 0.25 arcsecond. The distance to this star is a. 2 pc b. 0.5 ly c. 2 ly d. 4 pc e. 0.5 pc 14. On the main sequence, stars obtain their energy a. from chemical reactions. b. from gravitational contraction. c. by converting hydrogen to helium. d. by converting helium to carbon, nitrogen, and oxygen. e. from nuclear fission. 15. Star A has a radius that is 2 times larger than the radius of star B, and a surface temperature that is 2 times smaller than the surface temperature of star B. Therefore, star A is a. 4 times more luminous than star B. b. 16 times less luminous than star B. c. 16 times more luminous than star B. d. as luminous as star B.
16. A and B, two main sequence stars of the same spectral class, have apparent magnitudes of 17 and 12, respectively. If star A is 1 kpc away, what is the distance to star B? a. 10 pc. b. 100 pc. c. 10 kpc. d. 50 pc. e. 100 kpc. 17. Given that dark energy is vacuum energy, and that the densities of dark energy, dark matter and normal matter in the universe are currently ρ Λ= 6.7 × 10 g/cm −30 3, ρ DM= 2.4 × 10 g/cm −30 3and ρ Λ= 0.5 × 10 g/cm −30 3, what is the ratio of the density of dark energy at the time of the cosmic microwave background emission, to the current density of dark energy? a. 0.432 b. 2.31 c. 1 d. 2.5 e. 0.5 18. A type Ia supernova was observed in a galaxy with a redshift of 0.03. The supernova was determined to be 1.3 × 10 8pc away from Earth. Determine the Hubble time using this observation. a. 1.41× 10 10years b. 1.41× 10 10seconds c. 1.33× 10 9years d. 47.1 years e. 1.33× 10 9seconds
19.In a main sequence star, gravitational collapse is counteracted by: a. Radiation pressure
b. Heat c. Neutrinos
d. Electron degeneracy pressure e. Neutron degeneracy pressure
20.If the hydrogen alpha line of a star, normally 656.3 nm, is observed to be 662.5 nm, what is the star’s radial velocity relative to the Earth?
a. 2.83*106 m/s
b. -2.83*106 m/s
c. 0.00945 m/s d. -0.00945 m/s e. -2.83*103 m/s
21. Within M-type stars, heat transfer occurs primarily through: a. radiation b. conduction c. convection d. contraction e. collapse
22.If a 1.2 solar mass star shows a radial velocity variation with a period of 9.2 days and amplitude of 32 m/s , estimate the minimum mass of the companion:
a. 7.5*1026 kg
b. 1.2*1026 kg
c. 6.9*1027 kg
d. 5.1*1015 kg
e. 3.3*1027 kg
23. Calculate the planetary phase angle (counterclockwise from Earth, a = 1.0 AU) that a probe may correctly complete a Hohmann transfer orbit to Venus (a = 0.7 AU) a. 141 degrees b. 17.5 degrees c. 121 degrees d. 241 degrees e. 343 degrees
24.Calculate the blackbody equilibrium temperature of Mars. Take Mars’s albedo to be 0.25 and semimajor axis to be 1.5 AU
a. 300 K b. 212 K c. 161 K d. 228 K e. 260 K
25. Calculate the semimajor axis of a satellite orbiting the Earth with a velocity of 8.3 km/s at a distance of 300 km from the Earth’s surface.
a. 154 km b. 308 km c. 15800 km d. 7900 km e. 3950 km 26.On the night of December 23rd24th 2015, an occultation of a bright star by the moon will be visible from Britain to Japan. Given that the moon is in full phase on December 25th, which star does the moon occult?
b. Pollux (RA 7h 45m, Dec 28o 2’) c. Regulus (RA 10h 8m, Dec 11o 58’) d. Spica (RA 13h 25m, Dec 11o 14’) e. Antares (RA 16h 29, Dec 26o 26’) 27. A synodic day on Mars is 24 hours and 40 minutes. If one Martian year is 687 earthdays, which of the following is closest to a sidereal day on Mars? a. 23h 56m b. 24h 15m c. 24h 37m d. 24h 40m e. 24h 42m 28. Suppose at the equator, a star passes through the zenith at local noon on the summer solstice. What is the right ascension and declination of the star? a. 0h 0o b. 0h 90o c. 6h 0o d. 12h 0o e. 12h 90o 29. 40 light years away, an exoplanet orbits a star of 5 solar masses every 14 years. Assuming this system has an inclination of 90˚ as viewed from Earth, what is the projected diameter of the exoplanet’s orbit as viewed from Earth? a. 0.3” b. 0.8” c. 1.6” d. 2.5” e. 1.2” 30. A planet orbits a star with a projected semimajor axis of 0.24”. What is the necessary aperture size of a telescope than can resolve this orbit using 1000 nm light? a. 0.13 m b. 0.52 m c. 1.05 m d. 3.10 m e. 2.04 m