USAAO 2015 (Second Round)

SECONDROUND

2015

USA

ASTRONOMY & ASTROPHYSICS OLYMPIAD

Science Olympiad Blog

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]     (c) Calculate  the  actual  luminosity  and  radius  of  each  star.  [3]  

 

   

 

   

(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 (4.83−−0.88)/5 = 192Lsolar Lsolar = 100 From the Stefan-Boltzmann Law, L = 4πR2 σT 4 Finding the ratio with the sun, we get: 2  4  T L R Lsolar = Rsolar Tsolar  1/2
Tsolar 2 L R = = 4.6Rsolar Rsolar Lsolar T Problem 2 T = 3 days, K = 50 m/s, M = 1Msolar a3 T2 = M
3 2/3 = 0.041AU for the planet’s orbit. 365 For the star’s orbit: T = 2πr v r = T2πv = 1.38 ∗ 10−5 AU Relating the two, we have: m∗ r∗ = mp rp , so we have mp = rrp∗ m∗ Plugging in, we find mp = 3.36 ∗ 10−4 Msolar Problem 3 Solar rotation rate 24.5 days, MJ = 9.54 ∗ 10−4 Msolar , a = 5.2 AU, solar radius 695,000 km. L = Iω 2π Lsolar = 52 M R2 ∗ 24.5 = 5 ∗ 1010 For Jupiter, L = mrv r = 5.2 ∗ 149.6 ∗ 106 = 7.77 ∗ 108 km 2πr v = 5.22/3 = 1.13 ∗ 106 km/day. ∗365 −4 L = 9.54 ∗ 10 ∗ 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. 1

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 (Msolar −M )/5 = 59.9Lsolar Lsolar = 100 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 0 From Wein’s Law, λ = Tb . From the definition of Redshift, z = λ−λ 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 = 4πRp2 σTp4 L 2 Pin = A ∗ (1 − α) ∗ 4πD 2 = πRp (1 − α) Pin = Pout 2 4πR∗ σT∗4 4πRp2 σTp4 = πRp2 (1 − α) 4πD 2

2 4πR∗ σT∗4 4πD 2

R2

Tp4 = (1 − α)T∗4 4D∗2 q R∗ Tp = (1 − α)1/4 T∗ 2D Problem 9 7.2 micron pixels, f/10, D = 0.256 m. f/10, so focal length is 2.56 m. pixelsize ∗ 206265 Angular resolution is given by f ocallength 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. 2 µ , this corresponds to a velocity of Using the vis-viva equation, v2 − µr = − 2a 40,080 m/s at the Earth’s orbital distance.

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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 inqthe parking q orbit, the probe has a maximum velocity relative to GMe GMs the Sun of rorb + 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 Saturn’s orbital distance, and compute Saturn’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).

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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 1 proportional to twice the primary star’s radius. Dividing, we get rr21 = tt23 −t −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 2 luminosities:  4 L1 R1 T1 = 934 L2 = R2 T2 b. We use the provided chart to determine the orbital period of the binary v system (approximately 5.7 days) as well as the semi-amplitude (using ∆λ λ = c ). Using the period and velocity of each star, we determine the semimajor axes according to r = vT 2π , 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 m1 r1 = m2 r2 , 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 can1 dle, applying ∆M = 2.512 log F F2 . Similarly, using the luminosity of the system with the Sun’s absolute magnitude as a standard candle can provide an absolute magnitude for the system. Now that we have an apparent and absolute magnitude, we apply the distance modulus to get d = 5.94 pc. e. Applying the small angle formula, we get a maximum angular separation λ ∗ of 0.021”. The best possible resolution of the telescope is given by θ = 1.22 D 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. f. Examining the given plot, there appears to be a longer period variation

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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 m1 a1 = m2 a2 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.

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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?  o  a. 1​ o  b. 5​ o  c. 10​ o  d. 15​ o e. 20​   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  o​ (RA:5.27h, dec:45.98​ ) 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?  o  a. 1​

o  b. 5​ o  c. 10​ o  d. 15​ o e. 20​   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.   e. 4 times less luminous than 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 −30 g/cm 3 ,  ρ DM =  2.4  ×  10 −30 g/cm 3 and  ρ Λ = 0.5  ×  10 −30 g/cm 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 8 pc away from Earth. Determine the Hubble time using this  observation.  a. 1.41 × 10 10 years  b. 1.41 × 10 10 seconds  c. 1.33 × 10 9 years  d. 47.1 years  e. 1.33 × 10 9 seconds  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? 6​ a. 2.83*10​ m/s 6​ b. -2.83*10​m/s c. 0.00945 m/s d. -0.00945 m/s 3​ e. -2.83*10​ 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: 26​ a. 7.5*10​ kg 26​ b. 1.2*10​kg 27​ c. 6.9*10​ kg 15​ d. 5.1*10​kg 27​ e. 3.3*10​ 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?  o​ a. Aldebaran (RA 4h 37m, Dec 16​  31’) 

o ​ b. Pollux (RA 7h 45m, Dec 28​ 2’)  o​ c. Regulus (RA 10h 8m, Dec 11​  58’)  o​ d. Spica (RA 13h 25m, Dec ­11​  14’)  o​ e. Antares (RA 16h 29, Dec ­26​  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?  o a. 0h 0​   o b. 0h 90​   o c. 6h 0​    o d. 12h 0​    o e. 12h 90​   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 

     

 

USAAAO First Round 2015 Answers 1. C 2. D 3. C 4. B 5. C 6. B 7. D 8. A 9. B 10. C 11. B 12. B 13. D 14. C 15. E 16. B 17. C 18. A 19. A 20.B 21. C 22. A 23. D 24. B 25. D 26. A 27. C 28.C 29. C 30.B