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Celestial Celestial DistancesDistances

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Stellar DistancesStellar DistancesTo infer the luminosity, mass, and size of a star from observations (as in a celestial census), we need to know the distance to the starHow can we measure the great distances to stars? We use various techniques, useful at different scales, with each scale connecting to the next, like a ladder

On the Earth, lengths are specified in precise units such as the meterDistances within the solar system are determined by timing how long it takes radar signals to travel from the Earth to a planet or other body and then returnBeyond the solar system, …

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Parallax (1)Parallax (1)

To observers at points A and B, the tree at C appears in different directions This apparent displacement, or change in direction, of a remote object due to a change in vantage point is called parallax

The angle that lines AC and BC makes is also called parallax

The distance between A and B (length of line AB) is called the baseline

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Parallax (2)Parallax (2)How far away is the tree from each observer? One can use triangulation, a method for finding the distance to an inaccessible objectIf the baseline (B in this figure) and the parallax angle p are measured, then

the observers’ distances to the tree can be calculated using trigonometry

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Parallax for Stars (1)Parallax for Stars (1)The triangulation method can be applied to relatively nearby starsAs the Earth orbits the Sun, a nearby star appears to us to move back and forth against the background of distant stars This parallax can be used to find the distance d to the star

if the baseline and the parallax angle are known

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Parallax for Stars (2)Parallax for Stars (2)Since stellar distances are very large, the for has to be very large as wellFor a relatively nearby star, a sufficiently large baseline is the Earth-Sun distance, which is 1 AU

The farther the star, the smaller the angle pFor relatively far stars, extremely sensitive measurements of p are required

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The ParsecThe ParsecSince the parallax shifts of stars are very small, the arcsecond is used as the unit of the parallax angle

One arcsec (second of arc) is an angle of 1/3600 of a degreeThe parallax of the ball on the tip of a ballpoint pen viewed from across the length of a football field is about 1 arc second

With a baseline of 1 AU, how far way would a star have to be to have a parallax (p) of 1 arcsec?The answer is 206,265 AU, or 3.26 LY Astronomers take this number as another unit (besides the light year) for astronomical distances, called the parsec (abbreviated pc)

In other words, 1 parsec is the distance to a star that has a parallax of 1 second of arcThus, 1 pc = 206,265 AU = 3.26 LY

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More on the ParsecMore on the ParsecWhich unit to use to specify distances: the light year or the parsec?

Both are fine and are used by astronomers

For example, Proxima Centauri, the nearest star beyond the Sun, is about 4.3 LY, or 1.3 pc, away from us If the distance (D) of a star is in parsecs and its parallax (p) in arcseconds, then D and p are related by a simple formula: D = 1/pThus, a star with a parallax of 0.1 arcsec would be found at a distance of 10 pc, and another star with a parallax of 0.05 arcsec would be 20 pc away

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What about More Distant Stars?What about More Distant Stars?The triangulation method fails for stars farther than 1000 LY away

The baseline of 1 AU would be too small for sufficiently precise measurements of the parallax

Thus, completely new techniques are needed for more distant starsThe breakthrough in measuring the enormous distances came from the study of variable stars, or variables These are stars that vary in luminosity

Thus, their brightness changes with timeIn contrast, most stars are constant in their luminosity (at least within a percent or two)

Many variables change in luminosity on a regular cycle

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Cepheid Variable StarsCepheid Variable StarsOne of the two special types of variable stars used for measuring distances are the cepheids They are are large, yellow, pulsating stars named for the first-known one of the group, Delta Cephei

Its variability was discovered by English astronomer John Goodricke in 1784 It has a magnitude varying with a period of 5.4 days

time

Cepheid light curve

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Cepheid VariablesCepheid VariablesSeveral hundred cepheids have been found in our GalaxyMost have periods in the range of 3 to 50 days and luminosities in the range of 1,000 to 10,000 times greater than that of the SunPolaris, the North Star, is a cepheid variable

It used to vary by 0.1 magnitude every 4 daysMore recent measurements indicate that its pulsation is decreasing, which suggests that in the future it will no longer be a pulsating variable

Animation

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Cepheid Variables in NGC 3370 and M100 GalaxiesCepheid Variables in NGC 3370 and M100 Galaxies

The observations was taken by the Hubble Space TelescopeThe cepheid in NGC 3370 is in the center of a crowded region of stars and has a period of about 50 days

A cepheid in a very distant galaxy called M100

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RR Lyrae StarsRR Lyrae StarsAnother special special types of variable stars used for measuring distances are called the RR Lyrae variables

They are named for the star RR Lyrae, the best-known member of the group

They are more common than the cepheids, but less luminousTheir periods are always less than one day, and their changes in brightness are typically less than about a factor of 2From observations, astronomers have concluded that RR Lyrae variables all have nearly the same intrinsic luminosity, of about 50 times that of the Sun

Thus, they are like standard light bulbs

The RR Lyrae stars can be detected out to a distance of about 2 million LY

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Why A Cepheid Variable VariesWhy A Cepheid Variable VariesIts changes in color indicates a change in temperatureThe Doppler shift of its spectrum indicates a change in its sizeIts luminosity changes when its temperature and size change

In a normal star, the pressure and gravity In a normal star, the pressure and gravity balancebalance In a variable star, the pressure and gravity are In a variable star, the pressure and gravity are out of synchout of synch

cloud

pressure from hot gas

weight from gravity

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Period –Luminosity Relation (1)Period –Luminosity Relation (1)Studying photographs of the Magellanic Clouds, two small galaxies near ours, Henrietta Levitt in 1908 found 20 cepheids that were expected to be at roughly the same distance and discovered a relation between their luminosities and variation periods

The longer the period, the greater the luminosity

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 other clusters of stars) in another way

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Period –Luminosity Relation (2)Period –Luminosity Relation (2)Cepheids and their period-luminosity relation can be used to estimate distances out to over 60 million LY

under the assumption that all the cepheids obey the same period-luminosity relation

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Distances from H-R DiagramDistances from H-R DiagramVariables are rare and, therefore, cannot always be found near the a star of interestIf variables are not available, the H-R diagram may come to the rescue A detailed examination of a star’s spectrum can tell us its

spectral class/type (O, B, A, etc.)pressure and hence size (bigger stars have lower pressures)

Knowing the spectral class and size of a star can help us make an educated guess whether it is a main-sequence, giant, or supergiant starThis then allows us to pinpoint where the star is on the H-R diagram and establish its luminosityThe luminosity, with the apparent brightness of the star, finally leads to its distance

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SummarySummaryDistances to nearest stars can be measured using the parallax (triangulation) methodFor farther stars in our own and nearby galaxies, the distances can be determined using the RR Lyrae variables and the H-R diagramThe cepheids and their period-luminosity relation are useful for finding larger distances up to 60 million LY