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Astronomy 142 Recitation #2

Formulas to remember

Trigonometric parallax

Magnitudes and color indices (here all the ms and Ms are magnitudes)

Flux, distance and luminosity: .

Apparent magnitude and flux: This applies to any kind of magnitude, if the fluxes are measured in the same band as the corresponding magnitudes.

Absolute magnitude (= apparent magnitude for an object 10 parsecs away):

At any wavelength:

Bolometric:

Distance modulus: (see problem 1c, below)

Color index:

(see FA, p. 317, and also Homework #1)

Bolometric correction (get bolometric magnitude from V magnitude; see Figure 0):

Note that magnitudes are dimensionless.

Common (base 10) and natural (base e) logarithms:

By "log" we mean common logarithm base 10, and by "ln" natural logarithm.

BC

Figure 0: bolometric correction

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Planck blackbody function (power per unit area, bandwidth and solid angle):

; peak at

Note that our main textbook (FA) calls the Planck function . This is an eccentric choice of

symbols; the rest of the Universe uses and so shall we. The flux from a blackbody, emitted within a small bandwidth and a small solid angle :

.

Solid angle

Binary stars and their motions (here all the ms and Ms are masses; a is an orbital semi-major axis length; a = r for circular orbits)

Doppler effect:

Center of mass, circular orbits:

Kepler’s third law, any eccentricity:

Kepler’s third law, circular orbits:

Conservation of momentum:

Mass function:

Approximate Empirical scaling on the main sequence

R / M

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Workshop problems

1. Magnitudes and distances

a. The bright star Vega (α Lyrae; brightest star in the northern celestial hemisphere) has apparent magnitude 0.0 at all the usual visible wavelengths (U,B,V). What would its apparent magnitude at these wavelengths be, if Vega were moved a factor of five further away?

b. Vega’s parallax is 0.129 arcsec. What is its absolute V magnitude?

c. The Pleiades are a cluster of relatively young stars in the constellation Taurus. (The stars in a cluster that occupies a very small patch of sky, like the Pleiades, can all be assumed to lie approximately the same distance away from us.) Measurements of their apparent V magnitudes and B-V color indices appear in Figure 1. Given your results for Vega, how far away are the Pleiades?

d. What are the absolute V magnitudes of the bluest (smallest B-V) and reddest (largest B-V) stars in the Pleiades

e. What are the absolute bolometric magnitudes of the bluest and reddest stars in the Pleiades? (You will need to use the chart of bolometric corrections).

2. Eclipsing binary orientations

Suppose all binary stars consisted of two stars with radius orbiting each other with constant separation 1 AU.

a. From the viewpoint of one of these systems: what is the solid angle which, if observers lay within it, they could see the system eclipse? (Hint: recall the trig identity

.)

b. What is the solid angle of the whole sky?

c. What, therefore, is the fraction of binary stars in which we could detect eclipses? Compare this result to that noted in the two largest unbiased surveys that could detect binaries, those by the Kepler and Hipparcos satellites: 1.2% and 0.8% respectively (Prsa et al. 2010).

3. Binary stars and Kepler’s laws

a. Using Newton’s second law, prove Kepler’s third law for bodies in circular orbit about a much-more-massive object with mass M.

b. Derive the form of Kepler’s third law for two objects with equal mass m in circular orbits about their common center of mass.

4. Propagating Errors

Figure 1: apparent V magnitude as a function of color index B-V for the Pleiades (Stauffer et al. 1994).

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Observational astronomers often use the rule of thumb that at 1% change in brightness corresponds to a change of 0.01 magnitudes.

Using a photometer or CCD it is possible to measure absolute photometric measurements that are accurate to a precision of 0.01 mag. Given this error in measurement, what error results in the distance estimate to a star based on its apparent magnitude assuming that we know its absolute magnitude exactly?

5. Eclipsing binary color variations

On the left is a phase folded light curve of an eclipsing binary recently found in the 2MASS calibration database. On the right is a color vs magnitude diagram made using the points in the light curve. Each point is the color and magnitude at a different time. In the plot on the right bright points are high so the times in between the eclipses are on the top. During the primary eclipse the object is fainter and as seen on the color magnitude plot it is also redder. During secondary eclipse the object is bluer (and not as faint as in the primary eclipse).

a. What accounts for the color variations in the light curve?

b. Use the eclipsing binary simulator http://astro.unl.edu/naap/ebs/animations/ebs.html to explore how the shape of the light curve depends on the orientation angle. What do the shape of the eclipses tell you about the orientation?

5. SIMBAD

Go to the Astronomical Database:

http://simbad.harvard.edu/Simbad (SIMBAD Astronomical Database)

Click on:

Query by identifier:

phase

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Enter the name of a star

Here are some of my favorites.

Epsilon Eridani Beta Pic HD100546 Kepler 36

What time of year and in which hemispheres can you observe your star? Is there a listed proper motion or parallax? If there is a listed parallax how far away is the star? Is there any listed photometry? How bright is your star? Is it bright enough to see by eye or do you need a telescope to observe it? Is your star red or blue? What is your star’s spectral and luminosity class?

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Solutions

1. a. Flux decreases with the square of distance from the star:

.

We’re talking about the same L at two different distances; let’s call the near one and the further one . Then,

b. Vega’s distance, from its parallax, is , so

c. We may assume that the Pleiades with are just like Vega but lie at a different distance. Reading the V magnitudes of these stars off of Figure 1, we get . Calling the Pleiades 1 and Vega 2,

In other words, the distance modulus of the Pleiades is . All the Pleiades appear fainter than their absolute magnitudes by about 6 magnitudes.

(This distance is close to the best measurements, but not quite on the nose. The Hipparcos team measured the trig parallaxes of many Pleiades and got a distance of 135 parsecs, once they got their systematic errors under control.)

d. The bluest Pleiades have and the reddest have . Since their distance modulus is about 6.0 magnitudes, their absolute V magnitudes are , respectively.

e. The bluest Pleiades have and the reddest have . From Figure 0 we get respectively, so the bolometric magnitudes are

, respectively.

2. a. In class we showed that there would be a “grazing” eclipse if the orbital plane were inclined with respect to the line of sight by . Clearly there would also be a grazing eclipse for a tilt of

. So, in terms of the inclination angle with respect to an observer’s line of sight, i, an

eclipse corresponds to the range . Thus observers who see the system eclipse will occupy the solid angle

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Now use that identity, note that and , and use the small angle approximation whenever appropriate:

b. for the whole sky.

c. The fraction of observers – considered uniformly and randomly distributed over the sky -- who can see the eclipse is the same as the ratio of these two solid angles:

.

This, of course is the same fraction of eclipses that would be seen by one observer among a randomly-oriented collection of binary stars. The result is similar to what Kepler and Hipparcos actually got.

3. All you need is F = ma.

a. For example,

b. Each will orbit the center of mass, which is halfway between them:

4. Using a photometer or CCD it is possible to measure absolute photometric measurements that are accurate to a precision of 0.01 mag. Given this error in measurement, what error results in the distance estimate to a star based on its apparent magnitude (assuming that we know its absolute magnitude exactly)?

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5. a. The two stars have very different temperatures

b. The triangle shaped eclipses mean the system is orientated grazing. The face of one star only partly covers the other star at maximum eclipse.