Orbits for nine binaries and one linear solution
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The Schubart-like orbits in the collinear four-body problem are similar to those discovered numerically by Schubart in the collinear three-body problem. Schubart-like orbits are periodic solutions with exactly two binary collisions and one simultaneous binary collision per period. But not everyone agrees. The conversion of units is essential to recover a sense of the physical quantities such as planetary masses and years that we are familiar with.
Two body problem; formula for Doppler shift. An error analysis typical for frontier science. Two spectra are given for the spectroscopic binary Mizar A UMa. At the time of spectrum A, the spectrum lines are single. At the time of spectrum B, two days later, the lines are double. Their separation informs us of the radial velocity of the stars relative to each other. Estimate the uncertainty in your value: Your estimate of the uncertainty will be compared to the range of velocities measured by others.
The two lines in spectrum B are shifted nearly equally to the right and left of the single line in spectrum A. That suggests the two stars have nearly equal masses. Estimate the uncertainty in the ratio of masses, given your estimated uncertainty in radial velocity.
California Institute of Technology. The Sun is somewhat unusual in that it is a single star. About two thirds of all stars are binaries, that is, members of a binary pair of stars which orbit about each other under their mutual gravitational attraction.
The star orbits satisfy Kepler's third law extended to include comparable masses, see Appendix. The orbits of the stars around each other are extremely small compared to the distances of both stars from the Sun.
Only a few binaries are sufficiently close to us that the two stars can be photographed separately and the orbit can be measured completely.
For such binaries, one obtains the mass of each star. In many more cases, the two stars are so close together on the sky that they appear as a single dot on a photograph. We can recognize that the dot is a binary if we take its spectrum. The spectrum contains the light of both stars.
Differences in the radial velocity of the two stars show up as different Doppler shifts of the spectrum absorption lines. Because the stars orbit about each other, the Doppler shifts are seen to change from day to day. The graph shows a simple geometry with circular orbits. We see the orbit from the left. The Doppler shifts are identical when the stars are at positions A, but quite different at positions B.
Given sufficiently frequent spectra and information about the orientation of the orbital plane, one can deduce the star masses. The orbits involve unusually high radial velocities, high enough to allow a student's measurement of the radial velocity. Mizar A was the first spectroscopic binary to be discovered, in , because of the high velocities. Above are shown two spectra of Mizar A.
The two gray bands labeled A and B are the two spectra of interest. Each spectrum contains some darker vertical lines, referred to as spectrum lines. The wavelengths at which light is "missing" appear as vertical dark lines in the spectrum. In spectrum A, only one line appears at A. Thus the two stars have the same radial velocity. In spectrum B, taken two days later, the line from one star is Doppler shifted to longer wavelengths, the other to shorter wavelengths.
By comparing the distance between the split lines to the separation between the lines at While these measurement uncertainties appear in astronomical data, students should learn that that concerns about measurement uncertainties pervade physics, engineering, and all the physical sciences.
If physics students have little chance to measure things in the laboratory or in a city street, then data like these spectra may be used to help indoctrinate the students so that they will always ask about uncertainties before tackling a real physical problem. These spectra are quite old. More modern data can be much more accurate since this is a bright star. But the goal here is the students' experience with the limitations and uncertainties in measuring data.
The experience with uncertain data is an essential part of any scientist's and especially any astronomer's training. Even though error analysis can often be made much more precise than here, a judgment about possible errors based on experience is often the only possible way to inform other scientists about the reliability of new observations. The students themselves should measure the spectrum. Only a personal measurement of the line splitting, and an estimate of its accuracy, gives students a sense of the importance of worrying about accuracy.
If possible, a few copies of the spectrum should circulate through the class. Even if only a few students can measure, it is enough to establish the range of measurements. In any case, measurements should be recorded quietly, by each student, so that they will not influence each others' measurements. When all are finished, the accuracy of the measured line splittings can be established by comparing all the measurements and determining the spread in their values.
This practical result is much more important than the values given in the "solution". A mm-scale is the most likely measurement tool, but if a different scale inches? If a mm scale is used, there is a strong temptation to round off to whole mm, and that may give a systematic preference to one value. How do we know the wavelengths of spectrum lines?
The short white lines surrounding the spectra are made by a spark usually from an iron arc created within the telescope. These lines have wavelengths that are known accurately from the laboratory and are used to calibrate the lines appearing in the star spectrum. For the human eye, all the light contributing to the spectrum would have practically the same color. The Doppler shifts could not be perceived by the human eye as a change in color.
The masses of stars are the most fundamental property of stars, because the mass determines the star's surface temperature and size throughout its "lifetime" and the fate of the star at "death". Only few of all the stars have masses measured directly through binary orbits. A few very bright stars have their spectrum analyzed so carefully that the surface gravity and radius can be measured, which yields the mass.
For most stars, one matches the star's spectrum to the most similar spectrum of a star with a directly measured mass. Stars with relatively rare spectra have large uncertainties in M. For example, the companion to the black hole Cyg X-1 see problem 1. A stretch of the imagination. Interpretation of uncertain data. Demonstration of the existence of black holes. Assume a circular orbit.
This yields the minimum mass for the invisible object. To do this, first correct the modified Kepler's third law so that it is valid for an orbit of an unknown inclination, and then eliminate one of the velocities in terms of the other remembering action and reaction. Stars create their own gravity. If a star becomes smaller, the speed needed to escape from its surface increases. Eventually, when the star becomes sufficiently small, the escape speed exceeds the speed of light, c, and nothing can escape from the surface, not even radiation.
Such an object is referred to as a black hole. Actually the Schwarzschild radius was derived from general relativity by Karl Schwarzschild in The first good candidate as a black hole was the brightest x-ray source in the star constellation Cygnus, called Cyg X In , radio astronomers, detecting radio bursts from the same direction of the sky, located the x-ray emitting object much more precisely.
They showed that it is in the same position as a hot surface temperature about 3x10 4 o K and very large radius about 20 solar radii star which orbits about an invisible companion. The theory for the lives of stars section V says that the companion might be invisible because it is an extremely small star called a neutron star, with a radius of only 10 km, or because it is truly a black hole.
Neutron stars can have a mass no greater than 5Mo, for otherwise they collapse to become a black hole. Therefore, if we can show that the mass of the invisible object exceeds 5Mo, then the invisible companion must be a black hole. Several later problems deal with neutron stars, starting problem 3. Here the v are the actual orbital velocities, but v 2 is not observed. The left side is still 0. The mass derived for the invisible object is high enough for the object to be a black hole, even in the improbable case a.
There is some mass flow, but much more modest. The mass of the visible supergiant is uncertain in part because the spectrum of such a rare star is normally difficult to translate to a mass, in part because the visible star cannot be quite normal, situated so close to a sink for its gas and a powerful x-ray source. To summarize, the evidence for this black hole is very good, though by itself it does not convince all astronomers.
There are now other stellar objects with very good evidence for a black hole. The existence of stellar-mass black holes is no longer in doubt. How can we say that x-rays and radio waves are associated with stellar black holes when black holes do not radiate? The answer is that gas flows from the surface of the visible star, falls toward and begins to swirl around the black hole.
Indeed there is optical evidence for this gas. As it swirls ever closer to the black hole, it reaches temperatures on the order of 10 7 K, emitting x-rays. Only then does it fall into the black hole and disappear. Some of the x-rays, flickering on time scales less than one second, tell us that the gas falls into the black holes in chunks. Most students readily accept the notion of black holes. It is worth pointing out that astronomers are a conservative and skeptical crowd as is necessary to avoid mistakes in this frontier science.
Black holes with stellar masses were known theoretically for decades, but many astronomers doubted seriously that they exist in nature. The general acceptance of the reality of stellar-mass black holes had to wait until the necessary x-ray technology came along. Similarly, the reality of long-predicted neutron stars had to await the development of radio technology and the discovery of pulsars see problem 3. Since the derivation of Kepler's third law in the Appendix is rather cumbersome, it may be useful to provide to the class or have them derive the same law for circular orbits.
An outline of the proof appears in the Appendix. The relation is analogous to that found for some electrostatic problems. Used for our Galaxy, g r leads to the inference of much invisible, "dark" matter in our Galaxy. The stars that we see in the Milky Way constitute part of a large rotating disk of stars and gas that we call the Milky Way Galaxy or usually simply the Galaxy written with a capital G.
In the part of the sky containing the star constellation Andromeda, one can see with binoculars a spiral galaxy, called the Andromeda galaxy or M31, that probably looks similar to our Galaxy seen from a far. M31 is 2x10 6 light years from us. If the Sun were placed into the picture of the Andromeda galaxy, it would be near the outer part of the disk of stars.
Most of the stars are a few light years apart. Judging from the brightness of the center of the Andromeda galaxy, most of the stars of that galaxy are relatively near the center and distributed roughly spherically about the exact center. Although the center of our Galaxy is largely obscured from our vision by interstellar dust, we can tell that in our Galaxy also there is a roughly spherical distribution of stars around its center and distributed roughly spherically about the center.
If we take this observation literally, then the Sun is outside most of the mass distribution and we can treat all the mass of the Galaxy as being at the center. The center of our Galaxy lies at a distance, Do, of about 2. The assumptions allow use of the original Kepler's third law solar mass galaxy mass. The result, 10 11 Mo, is the reason that our Galaxy is often said to contain some 10 11 stars. But this is a very rough number, because most stars we know of have a mass less than 1Mo, so there must be more stars.
If we add up the mass of all the stars we see or infer behind the obscuring dust in our Galaxy, we obtain about 0. However, a first problem arises if we add up the stars in the central, roughly spherical region: So the assumption of a central mass comes into question. Instead, the circular velocity of the gas is observed to remain roughly constant from the Sun's distance outward as far as we can measure.
The observations mainly use the Doppler shift of the radio emission from neutral hydrogen at a wavelength of about 21cm. The constant velocity means that stars and gas further out than the Sun feel the gravity from more matter than the Sun does. At some distance, the velocity must decrease, but we cannot detect stars or gas there to measure the velocity.
In any case, there must be more mass than the mass in stars we see or infer behind dust. We speak of much "missing mass" or "dark matter". That means the dark matter must extend at least that far, well beyond the disk with its visible stars. One assumes that the dark matter is physically independent of the visible stars in the disk, and that it is distributed roughly spherically about the center.
The visible stars and gas, in this model, contribute rather little of the mass, but their rotation about the center, v r , traces the gravity g r caused by the dark matter. Let the rotation velocity v r be constant from Do to 5Do. What fraction of this mass is dark matter? Compared to luminous matter of 0. Estimates of the mass of the Galaxy are now about 7 at least 4 and perhaps 10 x10 11 Mo.
What is the dark matter? Perhaps there are unrecognized white dwarfs or black holes left over from former stars, or big planets or very cold gas. Indeed some star-like objects that we cannot detect by their radiation are now being detected because they gravitationally focus onto us the light from more distant stars. After a search of some three years, these newly detected star-like objects, probably white dwarfs with M 0. In cosmology, inflation theory and the search for the density which closes the universe has led to the suggestion that most of the missing mass in galaxies and clusters of galaxies, see problem 3.
This has led to a search for other forms of matter, such as neutrinos or axions. Most students have little trouble getting used to distances of stars we see in our sky, and to the scale of light years, because the Sun and the solar system still have some role to play as we learn about the stars. But students need time to adjust to the scale of our Galaxy. Photographs of the Milky Way do not help, because we see in the Milky Way only stars within some 3x10 3 light years from us, and these appear to be centered on the Sun and us.
The visible light from most of our Galaxy is obscured by dust. Maps based on radio observations appear very abstract to most students. But students can relate to photographs of other galaxies, such as the Andromeda galaxy, which probably resembles our Galaxy. And they can imagine the Sun places into the outer parts of the stellar disk of the Andromeda galaxy, and gradually revolving around its center.
It should be obvious here that there is no value in evaluating answers to this problem beyond one significant digit. It is important to point out that the uncertainty is not just due to measurement errors, but due to a real physical uncertainty in interpretation of the observations. This is valid only for spherical symmetry. If r equals the radius of the object, g r is the surface gravity. This gravitational problem can be used as a test that students have understood the analogous electrostatic problem.
A further stretch of the imagination. Some actual space data. Calculate the mass of the black hole at the center of the galaxy M84 if gas observed 0. Assume a distance of 5x10 7 light years.
In , quasars short for quasi-stellar objects were discovered to be among the most distant objects in the Universe, giving off hundreds of times as much radiation as do large galaxies, and yet, judging from their light variations, as small as light weeks. These strange properties led to the suggestion that the power must come from gravitational energy, specifically from material falling into a very massive and very tiny object, a black hole. A black-hole mass of 10 8 Mo was suggested, but this value seemed quite incomprehensible at that time.
Now we know that many galaxies contain very powerfully radiating centers, involving stellar masses up to 10 9 Mo. But are these centers really black holes? They might be merely dense accumulations of stars. In , the Hubble Space Telescope 2. This galaxy is a member of a "nearby" cluster of galaxies, located merely some 5x10 7 light years from us.
The galaxy is so nearby that the telescope could resolve light from merely 25 light years away from the center of M84 and measure the orbital velocity of the radiating gas.
The deduced mass of 3x10 8 Mo is not particularly unusual, but the resolved small distance from the center is.
If this mass consists of 3x10 8 stars distributed through a sphere of radius only 25 light years, then the stars are so densely packed 5x10 3 per cubic light year that they would have collided by now, and the result would be a black hole. The evidence for a very massive black hole at the center of the galaxy M84 is very good.