August 2010 Sky from the Keeble Observatory
We’ve seen over the last couple of months how astronomers can determine the distances to stars, as well as how we can use the spectra of those stars to determine their temperatures. Combining distance and temperature and the stars’ apparent brightness allows us to determine their physical size. This month we’ll find their masses.
We’ll need to look first at some characteristics of light from moving sources. When a light source is moving away from us, we detect it at a longer wavelength than when it is at rest with respect to our reference frame. We say that it is red shifted, because red light is at the long wavelength end of the visible spectrum. Conversely, if the source is approaching we say that it is blue shifted, because the wavelength we measure is shorter. This is the familiar Doppler effect, originally observed not for light but for sound, though the underlying physics for sound waves is different than for light. The amount of shift is directly proportional to the speed.
Another fact we’ll need is that most stars form as part of either binary systems, or they form with a set of planets. In either case, the orbital motions within that system are described by Newtonian gravitational theory. One consequence is that there is a harmonic relation between the period of an orbit and its size, and that relation tells us the combined mass of the system. The relation is known as Kepler’s Third Law, after Johannes Kepler, who discovered it empirically in trying to describe the orbits of planets in our own solar system. Newton showed that if the equations describing gravity take the form he suggested, that Kepler’s laws must be true.
So, here’s how we determine the masses for stars in binary systems:
For visual binaries we can sometimes trace out the elliptical paths on the sky of the two stars in the system. If we have already determined distance, we can then claim to know the size of the orbit, and over time we can measure the period, and hence calculate the total mass of the two stars combined. It’s complicated by our lack of knowledge of the plane of the orbits, since a tilted ellipse still looks like an ellipse, so we really only get a limit on those masses. Sometimes we get lucky … some binaries are eclipsing binaries (where the stars successively pass in front of one another), which means that the plane of the orbits is essentially along the line of sight, i.e. perpendicular to the imagined plane of the sky; which means that we can know the orientation.
If we apply our knowledge of the Doppler effect to these stars, we can also measure their line of sight velocities in their orbits. This is useful because the stars aren’t really orbiting one another, rather they are orbiting their common center of mass. Their orbital speeds are then inversely proportional to their individual masses, with the more massive star moving slower. So, we can not only know the total mass, we now have their ratio, which means that we can determine unambiguously their individual masses!
What we find is that main sequence stars of a given spectral type, i.e. of a specific temperature, are essentially the same. They have the same mass, same radius, same composition, same surface temperature, same intrinsic brightness, etc. This means that we can now determine distance by measuring the spectrum and the apparent brightness, which brings us full circle! We’ve captured a little light, and asked a lot of questions.
Next month we explore some consequences of knowing this.
Lunar phases for August: Last Quarter on the 3rd, at 12:59 am; New Moon on the 9th, at 11:08 pm; First Quarter on the 16th, at 2:14 pm; Full Moon on the 24th at 1:05 pm.
Jupiter rises before midnight, so predawn planet watchers will find it high to the west-southwest early in the month, about 45 degrees above the horizon. The rise time moves earlier as the month advances, so look for it lower and to the west-southwest by month’s end. A small telescope may enable you to find Uranus close by – about 3 degrees to the west of Jupiter early in the month, closing to about 1 degree later.
Early evening planet watchers get more to wonder at. Early in August Mars and Saturn are within a couple degrees, about 30 degrees above the west-southwest horizon. Venus is that brighter object below and to the right. Turning more to the West, Mercury should be bright about 14 degrees from the horizon as it emerges from twilight. By the end of the month Venus and Saturn will essentially trade places, with Saturn below and to the right of Mars and Venus, which are to the west-southwest. All will be closer to the horizon as the month goes on, and Mercury will disappear into the Sun’s glare.
Our overhead view at midmonth, about 2 hours after sunset, finds the bright star Vega in the constellation Lyra at zenith. The Milky Way arcs from north-northeast to south. That bright star to the east-northeast at about 65 degrees is Deneb, the “tail” of the swan in Cygnus, which appears to be flying along the plane of the Milky Way. To the southeast at 56 degrees altitude is Altair, in Aquila. These three bright “summer triangle” stars mark the season, as if the weather weren’t already doing that for us! Looking west from zenith we encounter an almost empty patch of sky, especially if you’re trying to see through the local glare of street lights. This direction is out of the plane of the Galaxy and is marked by relatively few nearby stars. The constellation Hercules makes a small irregular square about 20 degrees from zenith. The only truly bright star to the west is Arcturus, about 33 degrees above the horizon. Anticipating the coming season, we note that the Great Square of Pegasus is rising to the east, with the faint glow of the Andromeda galaxy clearing the northeast horizon.
Copyright 2010George Spagna