University of Michigan Invitational - MI — Astronomy

Astronomy · Division C · University of Michigan Invitational - MI · 2026

68 questions · 97 points · 50 min· 5 questions from the original omitted

Original test source

Test length: 50 Minutes

Section A (General)

Question 1

1 pt

If a planet's average distance from the sun is 20 AU, what is its orbital period in years?

A. 89.44B. 85.89C. 7.365D. 40.63

Question 2

1 pt

On the Visible Light Spectra, the wavelength of 440 nm - 490 nm corresponds to what color?

A. BlueB. VioletC. YellowD. Red

Question 3

1 pt

Identify the element using the following light spectra:

A. NeonB. XenonC. MercuryD. Argon

Question 4

1 pt

A star moving away from Earth will show spectral lines that are

A. BlueshiftedB. RedshiftedC. UnchangedD. broadened only

Question 5

1 pt

Which star type has the longest main-sequence lifetime?

A. O-typeB. B-typeC. G-typeD. M-type

Question 6

1 pt

The Crab Nebula (M1) is the remnant of

A. A planetary nebulaB. A Type Ia supernovaC. A core-collapse supernova observed in 1054 CED. A merging binary system

Question 7

1 pt

Which galaxy contains the star-forming region NGC 7023?

A. Large Magellanic CloudB. Small Magellanic CloudC. Milky WayD. Andromeda Galaxy

Question 8

1 pt

Type Ia supernovae occur when

A. A massive star collapsesB. A white dwarf accretes matter past the Chandrasekhar limitC. A protostar runs out of hydrogenD. A neutron star merges with a companion

Question 9

1 pt

A star with a parallax of 0.1 arcsec is at a distance of

A. 0.1 pcB. 1 pcC. 10 pcD. 100 pc

Question 10

1 pt

What does the Stefan-Boltzmann Law relate?

A. Mass and luminosityB. Luminosity and surface temperatureC. Radius and distanceD. Temperature and pressure

Question 11

1 pt

RR Lyrae stars are useful for measuring

A. distances to nearby galaxiesB. distances within globular clustersC. the age of star-forming regionsD. dark matter distribution

Question 13

1 pt

Which stage of stellar evolution immediately precedes the planetary nebula stage for stars?

A. Horizontal BranchB. Main SequenceC. AGB (Asymptotic Giant Branch)D. Red Clump

Question 14

1 pt

A star with an apparent magnitude of 2 and absolute magnitude of –3 is

A. closer than 10 pcB. farther than 10 pcC. Exactly 10 pc awayD. impossible to determine

Question 15

1 pt

What determines the final fate of a star (white dwarf vs neutron star vs black hole)?

A. TemperatureB. Rotation rateC. MetallicityD. Mass

Question 16

1 pt

A protostar becomes a true star when

A. Hydrogen fusion begins in the coreB. It forms an accretion diskC. It reaches the T Tauri phaseD. Its radius sharply expands

Question 17

1 pt

Which of the following is NOT a characteristic of a Cepheid variable?

A. Period–luminosity relationshipB. Pulsations caused by opacity changesC. Useful as standard candlesD. Formed only from low-mass stars

Question 18

1 pt

The luminosity of a star depends on:

A. Radius and temperatureB. Mass onlyC. Distance onlyD. Surface gravity

Question 19

1 pt

The equilibrium point between gas pressure and gravity inside a star is called

A. Jeans equilibriumB. Hydrostatic equilibriumC. Kepler’s balanceD. Roche stability

Question 20

1 pt

In binary star systems, the center of mass is located:

A. At the geometric midpointB. At the center of the larger starC. Closer to the more massive starD. Exactly between both stars

Question 21

1 pt

Inside T-Tauri Stars and other Pre-Main Sequence Stars, which particle is the primary cause of internal opacity?

A. HeB. H-C. e-D. He 2+

Question 22

1 pt

Stars collapsing down on the Hayashi Track start at roughly the same temperature. Why?

A. Protostars on the Hayashi Track are not in Hydrostatic EquilibriumB. He 2+ ions in the stellar interior keep opacity high, preventing much of the radiation from reaching the surfaceC. Radiative zone around the core allows radiation to escape efficientlyD. Stars on this track are mostly convective and gravitationally contracting

Question 23

1 pt

Roche-Lobe Overflow, a common phenomenon in close binary star systems, occurs at which lagrange point?

Question 24

1 pt

The Dynamical Timescale is a simplified mechanism to which of the following?

A. The amount of time it takes for open star clusters to dissipateB. The amount of time stars can radiate away any all stored thermal energyC. The amount of time it takes for a White Dwarf to exceed the Chandrasekhar LimitD. The amount of time for an object to expend all gravitational potential energy

Question 25

1 pt

For a constant density model sphere of radius R and temperature T collapsing by the Kelvin-Helmholtz Mechanism, what is the rate of pressure increase in the core proportional to?

A. ∝ R²T⁻¹B. ∝ T²R⁻¹C. ∝ T⁴R⁻¹D. ∝ TR

Question 26

1 pt

For orbital dynamics, what does the Reduced Mass Approach do?

A. Simplifies the three-body problem to a two-body problem, like a wide separated trinary systemB. Simplifies the interior of stellar atmospheres into layers of shells, each with a unique densityC. Reduces a two-body problem to a one-body problemD. Reduces the Cepheid pulsation frequencies in line to the fundamental frequency (n =1)

Question 27

1 pt

Which of the variable star types below have the largest period?

A. W VirginisB. ZZ CetiC. Type I CepheidsD. RR Lyrae

Question 28

1 pt

The pulsation period for a star of uniform density is proportion to what?

A. Π ∝ 1/R_StarB. ∝ 1/ρC. ∝ LuminosityD. ∝ 1/Luminosity

Question 29

2 pts

The radius of a white dwarf ____ as mass increases

Question 30

2 pts

Carbon stars generally form on what phase in stellar evolution?

Question 31

2 pts

What is the formal definition for the Eddington Luminosity?

Question 32

2 pts

If the pressure wave timescale becomes longer than the freefall timescale in a gravitational system, what occurs?

Section B (DSOs)

PassageFor questions B1b, B1c, B1d, B1e

The following questions pertain to The Bone (G359.13)

Question B1b

1 pt

What is significant about the fracture created by the pulsar collision?

Question B1c

1 pt

What wavelength are these “bone-like” structures most visible in?

Question B1d

1 pt

From Earth, what is the angular “diameter” of the bone in arcmin?

Question B1e

1 pt

What particles were accelerated at near-relativistic speeds, producing the electromagnetic radiation?

PassageFor questions B2a, B2b, B2c, B2d, B2e

The following questions pertain to WDJ181058.67+311940.94

Question B2a

1 pt

What is Earth’s distance to this DSO in ly?

Question B2b

2 pts

What is the mass ratio between the two stars in this system? (Both values m₂/m₁ or m₁/m₂ will be accepted) Must show work for full credit

Question B2c

1 pt

What image depicts this DSO? What is the significance of this star system?

Question B2d

1 pt

Once mass transfer of a certain element begins in this star system, it triggers a secondary detonation of the core of the primary star. What is the element?

Question B2e

1 pt

When this system goes supernovae, it is estimated to have a peak apparent magnitude of m_V = −16.2. Knowing the distance to this system, what will be the absolute magnitude?

PassageFor questions B3b, B3c, B3d, B3e

The following questions pertain to The Crab (M1 - Messier 1)

Question B3b

2 pts

The pulsar in the center of the nebula is powered by spindown luminosity. What is the basic principle of spindown luminosity?

Question B3c

1 pt

By an order of magnitude, what is the spindown luminosity of the Crab Pulsar in watts?

Question B3d

1 pt

Roughly what mass in solar masses did the progenitor star have?

Question B3e

2 pts

Why do lunar occultations of this nebula have scientific value?

PassageFor questions B4b, B4c, B4d, B4e

The following questions pertain to the Orion Molecular Cloud Complex

Question B4b

1 pt

The presence of shock bands in the cloud complex indicates what two stellar processes have/are occurring?

Question B4c

1 pt

The Orion Molecular Cloud Complex contains multiple regions of hot gas that are less dense than the surrounding interstellar medium. What are these regions called?

Question B4d

2 pts

The upper limit of the age of this DSO is 12 million years. Theoretically, what is the maximum mass (in solar masses) of a main sequence star in this DSO?

Question B4e

1 pt

About what spectral type does that solar mass from part d correspond to? Is that consistent with bright stars observed in the Orion Nebula?

PassageFor questions B5b, B5c, B5d, B5e

The following questions pertain to the HP Tauri

Question B5b

1 pt

By the classification of these stars, what track of evolution are they on according to an HR Diagram?

Question B5c

2 pts

Why are these stars expected to have a convective core? Why?

Question B5d

1 pt

What mechanism can be used to calculate the ages of these stars? What two variables does it involve?

Question B5e

1 pt

What is the apparent magnitude of HP Tauri (the brightest star in the trinary)?

Section C (Astrophysics) — 1) Simplified Binary Eclipse

PassageFor questions C1a, C1b, C1c, C1d, C1e, C1f

Let's assume we observe a binary star eclipse with stars S1 and S2. We will assume a simplified model with no limb darkening and both stars will be estimated as projected circles. Assume that our distance from the system ≫ the semi major axis of the system.

-	S1	S2
L_⊙	80.2	4.13

Question C1a

2 pts

What is the observed luminosity of the binary star when S2 eclipses S1 (across the "middle" of S1) in solar luminosities?

Question C1b

3 pts

Based on the total luminosity of the system without an eclipse, what is the change (∆m) in magnitude? (The sign on the change must be correct)

Question C1c

1 pt

Let's assume that S2 eclipses S1 offset from the middle of S1, with the disk of S2 still fully encompassed by S1. According to our model, would the change in luminosity be the same as in part a?

Question C1d

2 pts

Assume S1 now fully eclipses S2. What is the luminosity of the system during this eclipse in solar luminosities?

Question C1e

3 pts

What is the change (∆m) in magnitude for when S1 eclipses S2?

Question C1f

3 pts

For the S1-S2 system, the system has properties stellar masses (m₁, m₂), radii of the stars (r₁, r₂) and orbital radius a. Assuming a ≫ r₁ + r₂ and a circular orbit, derive t_E, the eclipse duration in terms of the given variables and relevant constants.

Section C (Astrophysics) — 2) 36 Ophiuchi System

FigureFor questions C2a, C2b, C2c, C2d, C2e

Question C2a

3 pts

Image F shows measured angular separations between components of the system. Determine the distance between 36 Ophiuchi A-B in AU.

Question C2b

1 pt

Orbital modelling shows that 36 Ophiuchi A and B have a semi-major axis of 13" (the orbits are not circular) and an orbital period of 546 years. What is the combined mass of the AB pair in solar masses?

Question C2c

2 pts

Orbital analysis shows that 36 Ophiuchi A is currently 2.567" from the center of mass. What is the individual mass of 36 Oph A and 36 Oph B in solar masses?

Question C2d

2 pts

Utilize the distance for the AB pair calculated in part a to determine at what distance in AU from 36 Oph A the gravitational forces from 36 Oph A and B would balance. What is this point called? Is it stable for an exoplanet?

Question C2e

4 pts

The 36 Oph AB pair system has an eccentricity of 0.916. Determine 36 Ophiuchi B's velocity/speed at periapsis relative to the barycenter for the AB pair in km/s

PassageFor questions C2f, C2g, C2h, C2i, C2j

From Earth, there are many binary or multiple star systems visible in the night sky. These systems can be put into two categories: I. Visual Systems (line of sight and not physically related) II. Physical Systems, which can be categorized even further into orbiting systems or a common proper motion system. (Both of which are influenced by gravity)

Each star's proper motion in celestial coordinates (Right Ascension and Declination) and radial velocity is also listed below. Use this information to answer the following questions.

36 Oph	Radial Velocity (km/s)	Proper Motion R.A. (mas/yr)	Proper Motion Declination (mas/yr)
AB	-0.3	-482.3	-1145.2
C	-0.04	−479.6	-1,124.3

Proper Motion Equation: v_t [m/s] = 1.454 · µ [mas/yr] · d [ly] Relative Escape Velocity: v_esc = √(2GM_Total/r) Space Velocity: v² = v_Radial² + v_t²

Question C2f

1 pt

What can be inferred about the C component and its motion?

Question C2g

2 pts

What is the magnitude of the relative proper motion of 36 Oph AB and 36 Oph C's in mas/yr?

Question C2h

1 pt

What is the magnitude of the relative radial velocity between 36 Oph AB and 36 Oph C in m/s?

Question C2i

5 pts

Using the condition v_relative < v_esc for a bound orbit, determine if 36 Oph C is an orbital component or Common-PM component of 36 Oph AB. 36 Ophiuchi has a mass of 0.72 M_⊙

Question C2j

1 pt

What is the period of 36 Ophiuchi C around the AB pair in years? (Write "Unbound" if it is not in a orbit)