Astronomy 102 Problem Set #3 Solutions

Problem #1: One day, while you're wandering through the streets of Lewisburg, you look up and notice two United States Air Force A-34 "Flying Weasel" fighter planes flying in formation directly overhead. You quickly measure that the angle between the line from you to the first plane and the line from you to the second plane is 4.7 degrees (that is, their angular separation is 4.7 degrees). Knowing as you do that when flying in formation, "Flying Weasels" are separated by 18 meters, you can calculate their altitude above your head. How high up are they?

Solution: This is a straightforward application of the Observer's Triangle relation for relating angular sizes, physical sizes and distances. If you haven't yet read the Special Web Page on this topic, do so now.

If you have read it, then you should be able to set up the appropriate triangle for this problem. We're given the angular size (4.7 degrees) and the physical size corresponding to this angle (18 meters), so use of the Observer's Triangle is straighforward:

So,

Note that the angular units (degrees) cancel out because they're the same top and bottom. The quantity 57.3 has units of degrees, so anytime you want to use this version of the Observer's Triangle you must measure your angular size in degrees.

Problem #2: Calculate the radius of a Main Sequence B0 star (which has a surface temperature of 3 x 104 K, and a luminosity of 1 x 103 Lo, where Lo means "solar luminosities").

Solution: Your class notes should provide you with the following relation between luminosity, temperature and radius for a spherical blackbody emitter:

where L is the luminosity, S is the surface area of the star, R is the radius of the star, T is the temperature, and sigma is the Stefan-Boltzmann constant. We can rearrange this equation and solve for R:

where x1/2 means "square root of x." Now we only need put in the input values from the problem and calculate R. However, we'll need to be careful about units. In particular, note that the luminosity in the problem is given in units of solar luminosities; that is, the luminosity of this star is 1000 times the luminosity of the Sun. So,

Just for kicks, we can calculate how many solar radii this is, since 1 solar radius = 6.96 x 108 m

Incidentally, this is actually a bit small for a B star. In putting together this problem, I think I chose a luminosity that was too small for a typical B star. The real size of a B star is a factor of three or so larger.

Problem #3: a) Calculate the distance to a star whose parallax is 0.012 arcseconds (NOTE: 1 arcsecond = 1/3600 th of a degree).
b) If this star has the same luminosity as the Sun, what is its flux (in W/m2) when viewed from the Earth?

Solution -- Part a):For this problem, you need to know what the parallax effect is, and more specifically, what is meant by the parallax of a star. You can learn about the former from your class notes, the book, or the Special Web Page on this topic. The latter is simply a definition, and can be found in all three places.

The parallax of a star is the angle through which a star appears to move when viewed from two positions separated by 1 Astronomical Unit (A.U.). It is the precise definition of the baseline (1 A.U.) which allows us to make a direct relationship between the angle and the distance to the star. With this information we can use the Observer's Triangle Relation to determine the distance.

For this problem, alpha is the parallax, or 0.012 arcseconds, w is the baseline, or 1 A.U., and R is what we're looking for, the distance to the star. Now before we can plug all of this into the Observer's Triangle relation, we need to pay attention to units. For this formulation of the Observer's Triangle Relation

alpha must have units of degrees. We know that 1 degree = 3600 arcseconds, so

Now the Observer's Triangle relation becomes

and therefore

where in the last step I just converted 1/5.81 x 10-8 into 1.72 x 107. In order to get the distance, we need only plug in our value for w

Note that our answer has units of A.U. because we used units of A.U. to express w. We didn't have to, but since we were given w in units of A.U., it was easier, and the answer is just as valid as it would be expressed in meter of parsecs.

We can convert this answer to meters or parsecs by multiplying by the appropriate conversions factor. Since 1 A.U. = 1.5 x 108 m

and since 1 pc = 2.06 x 105 A.U.

Note: Your book provides you with a shortcut for getting from the parallax of a star to its distance in parsecs:

This is a perfectly legitimate way of determining the distance from the parallax, but beware: you must use units of arcseconds for the parallax, and you must use units of parsecs for the distance. I'm not going to put this formula on your exam, and I just might be diabolical enough to ask you to compute the distance to a star whose parallax I give you in degrees. Using this shortcut formula is correct but dangerous. Make sure you understand how it works if you chose to use it.

Part b): We've seen in class that the flux from a luminous object is directly related to that object's luminosity and the distance between the observer and emitter as follows:

where f is the flux, L is the luminosity, and R is the distance between the observer and emitter. Note that R is not the radius of the star in this case.

We're told that the luminosity of this star is the same as that of the Sun, i.e., 3.8 x 1026 J/s, or W. We've just calculated the distance above, so let's put these quantities into our relation

Note that in order to get an answer in units of W/m2, you need to use a distance on meters. If you didn't calculate the distance in meters in the first part of this problem, you'll need to do some converting.