Chapter 19: Celestial Distances
How to Use a Cepheid to Measure Distance
(a) Find a cepheid variable star and measure its period. (b) Use the period- luminosity relation to calculate the star's luminosity. (c) Measure the star's apparent brightness. (d) Compare the luminosity with the apparent brightness to calculate the distance.
A light year measures
1 light-year = 9.46 × 1012 km 1 light-year is about 745 million times the diameter of Earth
length of light-second
3 × 108 m = 3 × 105 km
speed of light
3 × 108 m/s = 3 × 105 km/s
cepheid light curve
A graph that shows how the brightness of a variable star changes with time is called a light curve (Figure 19.9). The maximum is the point of the light curve where the star has its greatest brightness; the minimum is the point where it is faintest. If the light variations repeat themselves periodically, the interval between the two maxima is called the period of the star.
Parallax
An apparent shift in the position of an object when viewed from different locations Triangulation allows us to measure distances to inaccessible objects. By getting the angle to a tree from two different vantage points, we can calculate the properties of the triangle they make and thus the distance to the tree.
Distance to Stars
As Earth travels from one side of its orbit to the other, it graciously provides us with a baseline of 2 AU, or about 300 million kilometers. Although this is a much bigger baseline than the diameter of Earth, the stars are so far away that the resulting parallax shift is still not visible to the naked eye—not even for the closest stars
distance within solar system
Copernicus and Kepler established the relative distances of the planets -->couldn't establish absolute distances , to establish absolute distances, astronomers -->had to measure one distance in the solar system directly.--> measured estimate of Venus
calculating the diameter of the sun
EXAMPLE 19.2 pg 670
Luminosity Classes Stars
FIGURE 19.15 Stars of the same temperature (or spectral class) can fall into different luminosity classes on the Hertzsprung-Russell diagram. By studying details of the spectrum for each star, astronomers can determine which luminosity class they fall in (whether they are main-sequence stars, giant stars, or supergiant stars).
Measuring Parallaxes in Space
Hipparcos in 1989, which measured distances for thousands of stars out to about 300 light-years with an accuracy of 10 to 20% Gaia is expected to measure the position and distances to almost one billion stars with an accuracy of a few ten- millionths of an arcsecond. Gaia's distance limit will extend well beyond Hipparcos, studying stars out to 30,000 light-years (100 times farther than Hipparcos, covering nearly 1/3 of the galactic disk). Gaia will also be able to measure proper motions[2] for thousands of stars in the halo of the Milky Way—something that can only be done for the brightest stars right now. At the end of Gaia's mission, we will not only have a three-dimensional map of a large fraction of our own Milky Way Galaxy, but we will also have a strong link in the chain of cosmic distances that we are discussing in this chapter.
luminosity classes.
Ia: Brightest supergiants Ib: Less luminous supergiants II: Bright giants III: Giants IV: Subgiants (intermediate between giants and main-sequence stars) V: Main-sequence stars
If we can observe the spectrum of a star, we can estimate
If we can observe the spectrum of a star, we can estimate its distance from our understanding of the H-R diagram. As discussed in Analyzing Starlight, a detailed examination of a stellar spectrum allows astronomers to classify the star into one of the spectral types indicating surface temperature. (The types are O, B, A, F, G, K, M, L, T, and Y; each of these can be divided into numbered subgroups.) In general, however, the spectral type alone is not enough to allow us to estimate luminosity. BUTTT . A G2 star could be a main- sequence star with a luminosity of 1 LSun, or it could be a giant with a luminosity of 100 LSun, or even a supergiant with a still higher luminosity.
RR Lyrae stars
More common than the cepheids, but less luminous, thousands of these pulsating variables are known in our Galaxy. The periods of RR Lyrae stars are always less than 1 day, and their changes in brightness are typically less than about a factor of two.
variable stars
Most stars are constant in their luminosity, at least to within a percent or two. Like the Sun, they generate a steady flow of energy from their interiors However, some stars are seen to vary in brightness and, for this reason, are called variable stars. Many such stars vary on a regular cycle, like the flashing bulbs that decorate stores and homes during the winter holidays.
Period-Luminosity Relation for Cepheid Variables.
RR lyrae stars bottom left cepheids the rest
Parallax Diagram
Seen from opposite sides of Earth's orbit, a nearby star shifts position when compared to a pattern of more distant stars. Astronomers actually define parallax to be one-half the angle that a star shifts when seen from opposite sides of Earth's orbit (the angle labeled P in Figure 19.6). The reason for this definition is just that they prefer to deal with a baseline of 1 AU instead of 2 AU. As Earth revolves around the Sun, the direction in which we see a nearby star varies with respect to distant stars. We define the parallax of the nearby star to be one half of the total change in direction, and we usually measure it in arcseconds.
Distance Range of Celestial Measurement Methods
Table 19.1 page 682
using baseline surveyors and the moon
The Moon is the only object near enough that its distance can be found fairly accurately with measurements made without a telescope. Ptolemy determined the distance to the Moon correctly to within a few percent. He used the turning Earth itself as a baseline, measuring the position of the Moon relative to the stars at two different times of night.
With a baseline of one AU, how far away would a star have to be to have a parallax of 1 arcsecond?
The answer turns out to be 206,265 AU, or 3.26 light-years. This is equal to 3.1 × 1013 kilometers (in other words, 31 trillion kilometers).
naming stars
The brightest stars have names that derive from the ancients. Bayer- he assigned a Greek letter to the brightest stars, roughly in order of brightness. In the constellation of Orion, for example, Betelgeuse is the brightest star, so it got the first letter in the Greek alphabet—alpha—and is known as Alpha Orionis BUT only 24 letters in greek alph
baseline surveyors
The farther away an astronomical object lies, the longer the baseline has to be to give us a reasonable chance of making a measurement. Unfortunately, nearly all astronomical objects are very far away. To measure their distances requires a very large baseline and highly precise angular measurements.
pulsating variables & doppler effect
The lines in the spectrum shift toward the blue as the surface of the star moves toward us and then shift to the red as the surface shrinks back. As the star pulsates, it also changes its overall color, indicating that its temperature is also varying. And, most important for our purposes, the luminosity of the pulsating variable also changes in a regular way as it expands and contracts.
spectroscopic parallax
The method of determining a star's distance by comparing its apparent magnitude with its absolute magnitude, as estimated from its spectrum. The H-R diagram method allows astronomers to estimate distances to nearby stars, as well as some of the most distant stars in our Galaxy, but it is anchored by measurements of parallax. The distances measured using parallax are the gold standard for distances: they rely on no assumptions, only geometry. Once astronomers take a spectrum of a nearby star for which we also know the parallax, we know the luminosity that corresponds to that spectral type. Nearby stars thus serve as benchmarks for more distant stars because we can assume that two stars with identical spectra have the same intrinsic luminosity
How to use triangulation
The parallax is also the angle that lines AC and BC make—in mathematical terms, the angle subtended by the baseline. A knowledge of the angles at A and B and the length of the baseline, AB, allows the triangle ABC to be solved for any of its dimensions—say, the distance AC or BC. The solution could be reached by constructing a scale drawing or by using trigonometry to make a numerical calculation. If the tree were farther away, the whole triangle would be longer and skinnier, and the parallax angle would be smaller. Thus, we have the general rule that the smaller the parallax, the more distant the object we are measuring must be.
period-luminosity relation
The relation that describes how the luminosity of a Cepheid variable star is related to the period between peaks in its brightness; the longer the period, the more luminous the star.--> discovered by Henrietta Leavitt Leavitt discovered hundreds of variable stars in the Large Magellanic Cloud and Small Magellanic Cloud, two great star systems that are actually neighboring galaxies (although they were not known to be galaxies then). A small fraction of these variables were cepheids
cosmic distance ladder
The succession of methods by which astronomers determine the distances to celestial objects. Parallaxes are the foundation of all stellar distance estimates, spectroscopic methods use nearby stars to calibrate their H-R diagrams, and RR Lyrae and cepheid distance estimates are grounded in H-R diagram distance estimates (and even in a parallax measurement to a nearby cepheid, Delta Cephei). This chain of methods allows astronomers to push the limits when looking for even more distant stars. Recent work, for example, has used RR Lyrae stars to identify dim companion galaxies to our own Milky Way out at distances of 300,000 light-years. The H-R diagram method was recently used to identify the two most distant stars in the Galaxy: red giant stars way out in the halo of the Milky Way with distances of almost 1 million light- years
first successful detections of stellar parallax
Thomas Henderson, a Scottish astronomer working at the Cape of Good Hope, and Friedrich Struve in Russia independently measured the parallaxes of the stars 61 Cygni, Alpha Centauri, and Vega, respectively. Even the closest star, Alpha Centauri, showed a total displacement of only about 1.5 arcseconds during the course of a year.
parsec
We give this unit a special name, the parsec (pc)—derived from "the distance at which we have a parallax of one second." The distance (D) of a star in parsecs is just the reciprocal of its parallax (p) in arcseconds; D = 1/p Thus, a star with a parallax of 0.1 arcsecond would be found at a distance of 10 parsecs, and one with a parallax of 0.05 arcsecond would be 20 parsecs away. 1 parsec = 3.26 light-year, and 1 light-year = 0.31 parsec.
astronomical unit
When Earth and the Sun are closest, they are about 147.1 million kilometers apart; when Earth and the Sun are farthest, they are about 152.1 million kilometers apart. The average of these two distances is called the astronomical unit (AU). The length of 1 AU can be expressed in light travel time as 499.004854 light-seconds, or about 8.3 light-minutes. If we use the definition of the meter given previously, this is equivalent to 1 AU = 149,597,870,700 meters. These distances are, of course, given here to a much higher level of precision than is normally astronomical unit: AU = 1.50 × 1011 m = 1.50 × 108 km = 500 light-seconds
Proxima Centauri
an example of the most common type of star, and our most common type of stellar neighbor (as we saw in Stars: A Celestial Census.) Low-mass red M dwarfs make up about 70% of all stars and dominate the census of stars within 10 parsecs (33 light-years) of the Sun. if you wanted to see an M dwarf with your naked eye, you would be out of luck. These stars only produce a fraction of the Sun's light, and nearly all of them require a telescope to be detected.
Leavitt: brighter-appearing cepheids
brighter-appearing cepheids always have the longer periods of light variation. Thus, she reasoned, the period must be related to the luminosity of the stars. When Leavitt did this work, the distance to the Magellanic Clouds was not known, so she was only able to show that luminosity was related to period. She could not determine exactly what the relationship is. To define the period-luminosity relation with actual numbers (to calibrate it), astronomers first had to measure the actual distances to a few nearby cepheids in another way. (This was accomplished by finding cepheids associated in clusters with other stars whose distances could be estimated from their spectra, as discussed in the next section of this chapter.) But once the relation was thus defined, it could give us the distance to any cepheid, wherever it might be located
pulsating variables
cepheid and RR Lyrae variables, both of which are pulsating variable stars. Such a star actually changes its diameter with time—periodically expanding and contracting, as your chest does when you breathe.
Figure 19.8 H-R Diagram of Stars Measured by Gaia and Hipparcos
his plot includes 16,631 stars for which the parallaxes have an accuracy of 10% or better. The colors indicate the numbers of stars at each point of the diagram, with red corresponding to the largest number and blue to the lowest. Luminosity is plotted along the vertical axis, with luminosity increasing upward. An infrared color is plotted as a proxy for temperature, with temperature decreasing to the right. Most of the data points are distributed along the diagonal running from the top left corner (high luminosity, high temperature) to the bottom right (low temperature, low luminosity). These are main sequence stars. The large clump of data points above the main sequence on the right side of the diagram is composed of red giant stars
Why isnt this true: if earth orbited sun we would observe the parallax of the nearer stars against the background of more distant objects as we viewed the sky from different parts of Earth's orbit
how truly distant the stars were and how small the change in their positions therefore was, even with the entire orbit of Earth as a baseline. The problem was that they did not have tools to measure parallax shifts too small to be seen with the human eye. By the eighteenth century, when there was no longer serious doubt about Earth's revolution, it became clear that the stars must be extremely distant. Astronomers equipped with telescopes began to devise instruments capable of measuring the tiny shifts of nearby stars relative to the background of more distant (and thus unshifting) celestial objects.
the final meter
in terms of the velocity of light. Light in a vacuum can travel a distance of one meter in 1/299,792,458.6 second. Today, therefore, light travel time provides our basic unit of length. Put another way, a distance of one light-second (the amount of space light covers in one second) is defined to be 299,792,458.6 meters. That's almost 300 million meters that light covers in just one second; light really is very fast! we have defined the meter as a small fraction of the light-second
flamsteed star naming system:
in which the brighter stars eventually got a number in each constellation in order of their location in the sky or, more precisely, their right ascension. (The system of sky coordinates that includes right ascension was discussed in Earth, Moon, and Sky.) In this system, Betelgeuse is called 58 Orionis and 61 Cygni is the 61st star in the constellation of Cygnus, the swan --> NOW SPECIALIZED STAR CATALOGS
polaris : cepheid variable
is a cepheid variable that, for a long time, varied by one tenth of a magnitude, or by about 10% in visual luminosity, in a period of just under 4 days. Recent measurements indicate that the amount by which the brightness of Polaris changes is decreasing and that, sometime in the future, this star will no longer be a pulsating variable. This is just one more piece of evidence that stars really do evolve and change in fundamental ways as they age, and that being a cepheid variable represents a stage in the life of the star.
cepheid variables
large, yellow, pulsating stars named for the first-known star of the group, Delta Cephei (a whole class of stars is named after the constellation in which the first one happened to be found.) The star rises rather rapidly to maximum light and then falls more slowly to minimum light, taking a total of 5.4 days for one cycle. Most cepheids have periods in the range of 3 to 50 days and luminosities that are about 1000 to 10,000 times greater than that of the Sun. Their variations in luminosity range from a few percent to a factor of 10.
RR lyrae star clusters
occurring in any particular cluster all have about the same apparent brightness. Since stars in a cluster are all at approximately the same distance, it follows that RR Lyrae variables must all have nearly the same intrinsic luminosity, which turns out to be about 50 LSun. In this sense, RR Lyrae stars are a little bit like standard light bulbs and can also be used to obtain distances, particularly within our Galaxy
the first meter
one ten-millionth of the distance along Earth's surface from the equator to the pole
Magellanic Clouds
opportunity to study the behavior of variable stars independent of their distance. For all practical purposes, the Magellanic Clouds are so far away that astronomers can assume that all the stars in them are at roughly the same distance from us. If all the variable stars in the Magellanic Clouds are at roughly the same distance, then any difference in their apparent brightnesses must be caused by differences in their intrinsic luminosities.
Triangulation in humans
our depth perception fails for objects more than a few tens of meters away. In order to see the shift of an object a city block or more from you, your eyes would need to be spread apart a lot farther.
why is the problem with measure even the nearest stars?
parallax angles are usually only a fraction of a second of arc. Recall that one second of arc (arcsec) is an angle of only 1/3600 of a degree. A coin the size of a US quarter would appear to have a diameter of 1 arcsecond if you were viewing it from a distance of about 5 kilometers (3 miles). Think about how small an angle that is.
modern determination of solar system dimensions is
radar, a type of radio wave that can bounce off solid objects by timing how long a radar beam (traveling at the speed of light) takes to reach another world and return, we can measure the distance involved very accurately it is not possible to use radar to measure the distance to the Sun directly because the Sun does not reflect radar very efficiently. But we can measure the distance to many other solar system objects and use Kepler's laws to give us the distance to the Sun.
nearest star visible without a telescope from most of the United States
s the brightest appearing of all the stars, Sirius, which has a distance of a little more than 8 light-years. It too is a binary system, composed of a faint white dwarf orbiting a bluish-white, main-sequence star. It is an interesting coincidence of numbers that light reaches us from the Sun in about 8 minutes and from the next brightest star in the sky in about 8 years.
the nearest stars
tellar neighbors nearest the Sun are three stars in the constellation of Centaurus. To the unaided eye, the brightest of these three stars is Alpha Centauri, which is only 30○ from the south celestial pole and hence not visible from the mainland United States. Alpha Centauri itself is a binary star—two stars in mutual revolution—too close together to be distinguished without a telescope. These two stars are 4.4 light-years from us. Nearby is a third faint star, known as Proxima Centauri. Proxima, with a distance of 4.3 light-years, is slightly closer to us than the other two stars. If Proxima Centauri is part of a triple star system with the binary Alpha Centauri, as seems likely, then its orbital period may be longer than 500,000 years.
One of the most difficult things about precisely measuring the tiny angles of parallax shifts from Earth is
that you have to observe the stars through our planet's atmosphere
the redefinition of the Meter
the meter was redefined to equal 1,650,763.73 wavelengths of a particular atomic transition in the element krypton-86. The advantage of this redefinition is that anyone with a suitably equipped laboratory can reproduce a standard meter, without reference to any particular metal bar.
Edwin Hubble & Cepheids
using cepheids, when he observed them in nearby galaxies and discovered the expansion of the universe.
Distances from Spectral Types
variable stars have been for distance measurement, these stars are rare and are not found near all the objects to which we wish to measure distances--->turns out the H-R diagram can come to our rescue.
key reasons that measuring distances to the stars is such a struggle
variety of intrinsic luminosities apparent brightness --> we can calculate how far it is
what more can we learn from a stars spectrum?
we can detect pressure differences in stars from the details of the spectrum. This knowledge is very useful because giant stars are larger (and have lower pressures) than main-sequence stars, and supergiants are still larger than giants. If we look in detail at the spectrum of a star, we can determine whether it is a main-sequence star, a giant, or a supergiant.
