Lab6-Distance-Ladder
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Louisiana State University *
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1109
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Astronomy
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Apr 3, 2024
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Name: _________________________ Lab 6 –
The Distance Ladder Introduction
Distance is one of the most difficult things to measure in astronomy. You cannot see distance. You never know if you are looking at a low luminosity star nearby or a high luminosity star far away. In both cases, the star can appear to be equally bright. To deal with this, astronomers have developed a hierarchy of techniques for measuring greater and greater distances, called the Distance Ladder. In this lab, we will ex
plore the Stellar Parallax, Cepheid Variable Stars, and Hubble’s Law.
Parallax Parallax is a geometrical effect that can be used to obtain a direct measurement of the distance to an object. With current technology, parallax can only be used to measure distances out to about 500 parsecs, which is about 1.5 × 10
16
kilometers. This might seem like a great distance, but it is only about 7% of the way to the center of our Milky Way galaxy.
What is parallax? Questions: 1.
Predict what your finger would appear to do relative to the background if you were to put it about 3
–
4 inches from your face and close one eye at a time while watching it. 2.
Now go ahead and do this. Was your prediction correct? Comment. Notice how much your pencil appears to shift with respect to the distant objects. This is called parallax: the apparent shift of a foreground object with respect to background objects due only to a change in the observer's position. 3.
How the apparent motion of your finger would change if you moved your finger twice as far from your face? 4.
Now do this. Was your prediction correct? Comment. 5.
If you had amazing go-go-gadget stretchy arms, can you imagine a limit to how far you could move your finger and still see it appear to move? If so, how far away do you think that would be? (To get an idea of this distance, have someone far away from you hold up their finger.) 6.
What is it about our eyes that allows us to see this apparent motion?
The distance between the two observing positions is called the baseline. In the above exercise, the baseline is the distance between your eyes. Stellar Parallax In order to observe stellar parallax at all, astronomers must utilize the longest baseline that is available, namely the diameter of the earth's orbit: The stellar parallax angle, p, is defined by astronomers to be one-half the maximum change in position of the star relative to the background in one year, and so is the angle subtended by one Astronomical Unit (1 A.U. = average Earth-Sun distance) at the distance of the star. Because parallaxes are typically one arc-second or smaller, it is useful to calculate how large a distance this corresponds to. Since the distance to the stars is so very large, we can take the Earth-Sun line and replace it by the arc of the large circle of radius r centered at the star. (See dashed circle in Figure 5.) The circumference of this circle is 2πr. The length of the arc (1 A.U.) is in the same ratio to the circumference of the circle as the angle p is to 360 degrees. Expressed mathematically, this is 1 ?𝑈
2𝜋?
=
?
360°
Solving for r this may be expressed as ? =
360°
2𝜋?
× ?𝑈
Replacing 360 degrees by 1,296,000" (since there are 60" in 1' and 60' in 1o, there are 60 x 60 x 360 = 1,296,000" in 360 degrees) and dividing by 2π, we obtain
? =
206265"
?
?𝑈
If the parallax angle is equal to 1" of arc, the distance is thus 206,265 A.U. This distance = 206,265 A.U. is given a new name, 1 parsec, meaning the distance corresponding to a parallax of 1 second of arc, using a baseline of 1 A.U. When distance is expressed in parsecs (abbreviated pc) and parallax angle is expressed in seconds of arc, the previous equation becomes simply ?(𝑖? ??) =
1
?(𝑖? ??????? ?? 𝑎??)
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Another unit of distance commonly used in astronomy is the distance traveled by light in one year at the speed of 3.00 x 10
8
m/sec. This distance, called the light-year (L.Y.), is equal to 9.46 x 10
12
km, or 6.324 x 10
4 A.U. It is easy to show that 1 pc = 3.26 L.Y. The distance to a star in L.Y. is given by ?(𝑖? 𝑙?) =
3.26
?(𝑖? ??????? ?? 𝑎??)
Questions: 7.
In the table below are several stars. Complete the table by filling in the missing quantities, applying the distance formulas derived above. Star Name α
-Centauri Sirius Pollux Betelgeuse Distance (pc) 160 pc Distance (ly) 4.3 ly 35 ly Parallax (“)
0.38”
8.
The photographs in Figure 6 represent the same section of the sky taken at an interval of six months. If you will examine the pictures carefully you will note that all of the stars in the two pictures are in the same positions except one, which is shifted. Identify which of the stars has moved and circle it on the two pictures. 9.
The star that moves in Figure 6 shifts by about 18 mm. Calculate the amount of motion. Total parallax = __________”
Stellar parallax = __________”
Distance = __________pc = __________ly 10.
Which of the stars from the above table might this star be?
11.
In Figure 6, Set A has a 20 mm shift, Set B has an 18 mm shift, and Set C has a 12 mm shift. Assuming the scale to be the same for all three sets of photographs, which set indicates the most distant star? Explain. 12.
Which set of photographs indicates the closest star? Explain. Cepheid Variable Method If the luminosity, L, of the star is known, its distance, D, can be calculated by measuring its brightness, B: 𝐷 =
√
?
4𝜋?
For distant stars, you often don’t
know the luminosity, however, so you cannot use the above equation to calculate its distance. An object of known luminosity is called a standard candle
. Most stars are not standard candles —
their luminosities are not known and consequently their distances cannot be easily calculated. However, some special types of variable and exploding stars do have known, standard luminosities. Consequently, if you can identify a star as being one of these special types, you know its luminosity. Then you only have to measure its brightness to be able to compute its distance. Instead of brightness and luminosity, astronomers use apparent magnitude, m, for brightness and absolute magnitude, M, for luminosity. The above equation giving D as a function of B and L can be rewritten as an equation giving D as a function of m and M: D = 0.01 kpc x 1.585
(m-M)
Cepheid stars are a type of variable star. Their outer layers expand and contract over and over. They grow brighter as they expand and fainter as they contract. A typical light curve will look something like this Distances have been measured to nearby Cepheid stars in our galaxy using parallax techniques. These distances, in combination with measurements of the stars' average apparent magnitudes (think average brightness), made possible the calculation of their average absolute magnitudes (think average luminosities). It turned out that all Cepheid stars have average absolute magnitudes that are related to their periods in the following way. This can be written as a mathematical relation (note that this is a base 10 logarithm): ? ≈ −1.43 − [2.81 × log (
𝑃
1 ?𝑎?
)]
This means you only need a measurement of the period in days to get the absolute magnitude of a star, and when you add in the apparent magnitude, you can find the distance to that star.
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Here is a light curve of Delta Cephei, a Cepheid variable star. Questions: 13.
What is your estimate of the period of this star in days? 14.
Convert this period into an absolute magnitude using the equation given above. 15.
Delta Cephei has an apparent magnitude (m) of 4.07. Use this and the absolute magnitude (M) that you just calculated to estimate the distance to this star in kiloparsecs. (D = 0.01 x 1.585
(m-M)
kpc
)
16.
We observe a Cepheid in a galaxy. It has a period of 10 days. It has an apparent magnitude of 18.5. What is the absolute magnitude of this Cepheid?
17.
How far away is the galaxy? The Hubble Law The speed at which an object appears to be moving toward or away from us can be measured by how blue or redshifted the object's spectrum is V = cz
Where c is the speed of light (3.0 x 10
5
km/s), and z is the redshift, given by the equation ? =
∆𝜆
𝜆
?𝑚𝑖????
=
(𝜆
𝑜𝑏???𝑣??
− 𝜆
?𝑚𝑖????
)
𝜆
?𝑚𝑖????
Hubble found that except for very nearby galaxies, which are gravitationally bound to us, all galaxies appear to be moving away from us. Furthermore, the farther away a galaxy is from us, the faster it appears to be moving from us. This is Hubble’s Law.
The slope of this line is a constant which we now call the Hubble constant, with a value of about 70 km/s/Mpc, and gives us a relation that we can use to find the distance D = v/H
0
Where H
0
is the Hubble constant. Questions: 18.
Fill in the Table below using the equations given to you on this page, if the emitted wavelength is 656 nm Star Observed Absorption Line Redshift (z) Velocity (km/s) Distance to star (Mpc) A 660 nm B 713 nm C 777 nm D 824 nm E 936 nm