[ kd ]
hwk01
Ast 4001, 2015 Sept 8
Homework set 1 -- Practice exercises concerning distances and absolute magnitudes
Elementary-level calculations, aimed at becoming familiar with the topic. Many research
astronomers routinely do calculations like these almost every day. Turn in your solutions
to problems 5 and 6 on Thursday September 17. The others are for practice, and invent
similar problems yourself, because the only way to learn a language, a skill, instrument,
or technique is to practice. Besides, questions like these often appear on exams.
Moreover: Always ask yourself “does this answer make sense?” Everyone makes calculative
errors, but we should always notice if a number is ridiculously small or large. .
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1. On a mediocre night, most research telescopes have angular resolution about 1 arcsec,
limited by turbulence in Earth’s atmosphere. At what distance would the Sun have an
apparent angular diameter of 1 arcsec? Express the answer in AU, pc, and meters.
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2. A particular star cluster has a diameter of 7 pc. If its distance is 2300 pc, calculate the
cluster’s apparent angular size expressed in degrees, arcmin, and arcsec.
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3. At maximum brightness, a typical supernova might have visual absolute magnitude
M = about 18.
(a) Compare its visual-wavelength luminosity to the sun, L (SN) / L (sun).
(b) At what distance would the supernova appear as bright as Sirius, m =  1.4 ?
Before looking up a formula, try to figure this out for yourself based only
on the definition of absolute magnitude M .
(c) At what distance would it appear as bright as the full Moon?
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4. Look up information about Proxima Centauri, a.k.a. alpha Centauri C, the most famous
red dwarf star. Find out its absolute visual magnitude M and its approximate distance
from the brighter central star, alpha Centauri A. Estimate Proxima Centauri’s apparent
magnitude as seen from alpha Cen A, and alpha Cen's magnitude seen from Proxima.
Rough answers are sufficient, if you understand which factors cause the uncertainty.
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5. Estimate the bolometric absolute magnitude Mbol of a 100-watt light bulb.
Turn in your solution on Thursday Sept. 17, with a clear explanation of your reasoning.
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----- one more problem is on the next page or the other side of this sheet -----
4001hwk01 - p2
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6. Turn in your solution to this problem on Thursday Sept. 17. Explain your reasoning and
calculations clearly ... and please write legibly!
The figure shows observations of a visual binary star with period = 60 years. If we take random
observational errors into account, the data indicate a circle shown on the right. But the true orbit
in space cannot be a circle, because the primary star is far off-center. The orbit must be a highly
eccentric ellipse seen from an oblique angle. The apparent diameter of the circle is 1.8 arcsec.
A bit of practical advice: If you have a picture that you can display on a computer, the
best way to measure something in it is to use the computer. Display the image with any
good graphics software, and find a function that gives cursor’s pixel coordinates. (For
example, some graphics routines display the cursor x and y at the lower left or lower
right.) Remember that screen-capture or “print screen” on the keyboard makes a
graphics file of whatever appears on your display screen, preferably zoomed in. An
alternative method is to use a copier to make a greatly magnified image on paper.
(a) Use the right-hand figure to estimate the orbit’s eccentricity . Explain.
(b) Estimate the orbit’s semi-major axis a expressed in arcsec, i.e., the value
that we would see if the orbit were perpendicular to our line of sight.
(c) Estimate the orbit inclination angle i that makes this orbit appear circular.
Inclination i is defined to be zero if the orbit plane is perpendicular to our
line of sight, and 90º if we view it edge-on.
(d) This binary star system is at a distance of 15 pc from us. Use your result
in part (b) to calculate the true linear value of a expressed in AU.
(e) Deduce the total mass of the system, expressed in units of M sun .
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