Applying HR Diagrams: Spectroscopic Parallax & Stellar Sizes Version: 1.0 Author: Sean S. Lindsay History: Written October 2019; Last Modified: 28 October 2019 Goal of the Lab The goal of this lab is to extend our understanding of the properties of stars and how they relate to the HR diagram. In this lab, you will explore stellar luminosity classes, the luminosity-radius-temperature relationship, and the method of determining distances to stars called spectroscopic parallax. This will extend the usefulness of HR diagrams and further develop the connection between HR diagrams and understanding the evolutionary life cycles of stars. By the end of this lab, students are expected to know:

1. The Morgan-Keenan (MK) Luminosity Classes of stars 2. The distance determination method called Spectroscopic Parallax that is based on the HR diagram 3. The connection between stars luminosity, temperature, and radius via the luminosity-temperature-

radius 4. Maybe Include: Main-sequence turn-off and stellar lifetimes.

Tools used in this lab:

• A online table of Spectral Types temperatures: • A ruler/straight-edge

1. Introduction: The Tale of Six Stars In this lab, you will gain practical experience using the HR diagrams you explored in the previous lab. The goal is to learn how astronomers use HR diagram to connect properties of stars, and how we can use the HR diagram to determine the absolute magnitude of stars, which can, in turn, be used to calculate the distances to stars that are too far away for stellar parallax to be measured with our current technology. For the connecting properties of stars, you will see how the luminosity (the HR diagram y-axis) and temperature (the HR diagram x-axis) can be used to determine the radius of the star. As for using an HR diagram as a method to determine distances, we need to fill in a bit more background about stars (See Section).

The ultimate goal of this lab is simple. Complete Table 1 on the student worksheet.

Fig. 1: A Hertzsprung-Russell Diagram. In this scatterplot of luminosity versus temperature, stars cluster into three primary regions: The giant region, the main sequence, and the white dwarf region

Giants

Main Sequence

White Dwarfs

That task, however, sounds rather dry, so let’s give it some meaning. In this lab, you are going to write the tale of six real stars. You will:

1. Determine Observable Properties Using Stellarium: Use their observable properties found via Stellarium to determine their apparent, B – V color-index, and spectral class.

2. Determine the Temperature using Spectral Class: Instead of going through the University of Nebraska, Lincoln “Blackbody Curves and Filters Explorer,” to translate B – V to temperature, you will use a look-up table that connects the star’s classification to its temperature.

3. Determine the Absolute Magnitude & Luminosity from the HR diagram: You will then use that information to plot the stars on an HR diagram, and use the HR diagram to determine their absolute magnitudes, which is equivalent to the stars’ luminosities.

• There is a new trick here that we did not consider in the previous lab. Recall that on the HR diagram, we had the evolved stars in the “Giants” and “White Dwarfs” regions of the HR diagram. If I know the spectral class of my star to an F5 star, is that star an F5 main sequence star, or is it an F5 giant? Are all giant stars the same type of giants?

• Being the dead cores of low-mass stars, white dwarfs don’t have the spectral lines that main sequence and giant stars have, so how do we put those on the HR diagram? In this case, I you will use the B – V color index, and I will give you the temperature.

4. Determine the Distance to the Star with the Distance Modulus: You now know each star’s apparent and absolute magnitude. You can use that information to calculate the distance modulus m – M, which can be used to calculate the distance to the star.

• This method of determining distance is used for stars that are too far away to have measurable stellar parallax

• This method is called spectroscopic parallax. This is a horrendous misnomer because it has nothing to do with the phenomenon known as parallax.

5. Determine the Radius of the Star: The astronomer’s version of the Stefan-Boltzmann Law (a thermal radiation law explored a few labs ago), connects the star’s luminosity to the star’s size and temperature. This gives what is known as the luminosity-radius-temperature relationship. You will use each star’s temperature and luminosity to calculate the physical size of the star.

This is the power of the HR diagram. With just a few observable quantities, you can start accessing physical properties of the stars. This allows astronomers to understand the properties of stars and start making further connections between say the mass of stars and their temperature, or the luminosity of the star and the temperature which can be used to determine how long stars will live on the main sequence. I hope you appreciate just how powerful of a tool you are playing with here. A little bit of light from the stars coupled with a good bit of know-how of how light and nature works allows you to unlock the essential nature of the stars themselves.

2. the Morgan-Kennan (MK) Luminosity Classes Before we can get to work, we need to resolve the problem of a single spectral class, e.g., F5 could be a low-luminosity main sequence star or a high-luminosity giant star. For a single spectral class, there are multiple luminosity, or equivalently absolute magnitudes, that the star could truly have. Perhaps we should take a closer look at where stars fall on the HR diagram. Figure 2 shows an HR diagram for 22,000 stars in the Hipparcos1 Catalog.

Figure 2. The HR Diagram for 22,000 stars in the HR diagram. The large number of stars reveals that “giant” stars do not all fall in the same region. The majority of giant stars fall in the “Giants,” but the other large stars can be subdivided into three other regions: subgiants, bright giants, and supergiants. This is the basis of the Morgan-Keenan Luminosity Classification scheme.

The large number of stars reveals that “giant” stars do not all fall in the same region. There are specific regions in the HR diagram where stars cluster. This provides the basis for the Morgan-Keenan (MK) Classification. Based on where stars cluster on the HR diagram, the MK Classification identifies FIVE separate luminosity classes (plus white dwarfs). Each of these luminosity classes is designated with a Roman numeral. The main sequence stars are giving the luminosity class of V (“five”). The luminosity classes are:

• Luminosity Class, V - Main sequence stars • Luminosity Class, IV – Subgiant stars • Luminosity Class, III – Giant stars • Luminosity Class, II – Bright Giant stars • Luminosity Class, I – Supergiant stars

o Luminosity Class I is further supdivided into: Ia – Luminous Supergiant stars Ib – Less Luminous Supergiant stars

1 The Hipparcos satellite was launched by the European Space Agency (ESA) in 1989 and operated until 1993. It used stellar parallax to determine with distances to 118,200 stars with unprecedented accuracy. The ESA launched a follow-up mission called Gaian 2013, which is determining the parallax to approximately 1 billion stars.

The luminosity classes extend the Harvard Spectral Classification (O B A F G K M) + (0 – 9) explored in the previous lab. Sun is a spectral class G2 star. It is also a main sequence star, which is luminosity class V. So, the full spectral classification for the Sun is G2 V. As the Sun ends its life on the main sequence, it will cool down and become a red giant star. It will become a K- or M-type star and first change to luminosity class IV star (subgiant on the subgiant branch of stellar evolution) and eventually become a luminosity class III star (giant on the red giant branch of stellar evolution).

The spectral classes only account for the star’s spectral characteristics based on its temperature. The luminosity classes account for the star’s spectral characteristics based on its temperature, and the surface gravity effects on a star’s spectrum. The physics is a bit complicated, so the details are left out here, but as a star expands to become a giant or supergiant star, the surface gravity becomes smaller. The star is the same mass, but now the surface is much farther away, so the surface gravity is lower. The lower surface gravity manifests itself in the spectrum of a star in a few ways. Primarily, larger stars with lower surface gravity have:

• Narrower spectral lines due to decreased pressure broadening • Subtle changes in the ratio of line strengths.

The narrower spectral lines are the easiest to observe and the most prevalent. Main sequence stars have the highest surface gravity, and hence the highest surface pressure and the most pressure broadening of lines. In general, supergiant stars have the lowest surface gravity, and hence the smallest surface pressure and the least pressure broadening of lines. You can see an example of the difference in spectral lines for G-type star based on luminosity class in Fig. 3.

Figure 3. The spectra of five G5 spectral class stars. From top to bottom, the luminosity classes are I (supergiants) to V (main sequence). The G5 V star has the widest spectral lines while the G5 I star has the narrowest. This is most apparent with the neutral Mg line near 520 nm and the 𝐻𝛼 hydrogen Balmer line at 656.3 nm. There are also some observable differences in the strength of some of the other spectral absorption lines.

Luminosity Class I – Narrowest Lines

Luminosity Class V – Broadest Lines Credit: S. Lindsay (Spectra files from A.J. Pickles, PASP 110,863, 1998)

Measuring the width of spectral lines, therefore, gives astronomers a way to determine the luminosity class of stars. The process and details of this are difficult, so other than one question about identifying the supergiant versus main sequence star, the luminosity classes will be given to you in the lab.

Name:

Lab Instructor:

Lab Date & Time:

Lab Task 1: Plotting 6 Stars on an HR Diagram The overall goal of the lab is simple: Complete Tables 1 & 2, plot the stars on the provided HR diagram, and then calculate the radii of the stars. You will use Stellarium to “observe” three of the six stars. The stars you are observing include: Three F-type stars, a B-type supergiant, a M-type star, and a white dwarf star.

1. Open Stellarium. Use the Search function look up the stars and navigate to them. Copy the relevant observable data. The Stellarium “Search Window” is on the tool bar that pops up when you move your cursor to the left-hand-side of the Stellarium Window. Write down the apparent magnitude, “Magnitude: “ NOT the Absolute magnitude

Table 1. What you can observe at the telescope

Star Apparent

magnitude, m B – V Color

Index Spectral

Class Luminosity Class

Wezen Ia – Supergiant

Iota Piscium (Iota Psc) V – Main sequence

Caph III - Giant

S Doradus (V* S Dor) 8.6 0.11 B8 Iae – Extremely Luminous Supergiant

Gliese 581 10.57 1.55 M3 V – Main sequence

Sirius B 8.44 -0.10 --- White Dwarf

2. Complete Table 2 as you work through the questions in this lab. For Sirius B, it may help to draw an “average” line through the White Dwarf star region.

Table 2. Properties of the star

Star Absolute mag., M

Luminosity (LSun)

Temp. (Kelvin)

Temp. (TSun)

Distance (parsecs)

Radius (RSun)

Wezen

Iota Piscium (Iota Psc)

Caph

S Doradus (V* S Dor) -10.0 8,500 K

Gliese 581

Sirius B 25,000 K

3. Using the B – V color-index and Luminosity Class from Table 1, plot and label the six stars on the HR Diagram.

4. Use a ruler to determine the absolute magnitude, M for each of the six stars. Record the absolute magnitudes on Table 2. Due to the special, extremely rare, luminosity class for S Doradus, the absolute magnitude has been provided for you. Try to be as accurate and precise as possible!

5. Use the equation to convert absolute magnitude into luminosity in solar units. This is the same equation you used in the previous lab. Record the luminosities on Table 2.

𝐿∗ = 10(.*(*.,*-.)

6. Use the Excel Spreadsheet “Temperatures of Spectral Classes” provided on the UTK Astronomy Lab website to determine the temperature of the stars based on their spectral class. The temperatures of Sirius B, the white dwarf, and the extremely luminous supergiant S Doradus are provided for you.

a. Record the temperature in Kelvin on Table 2.

b. Convert the temperature to number of solar temperatures. The temperature of the Sun is 5,800 K, so you simply just need to divide the star’s temperature by 5,800 K.

You might not yet realize it, but we have accomplished something remarkable here. Without knowing the distance to the star, we have gotten an estimate of each star’s absolute magnitude and luminosity. This is one of the powerful aspects of the HR diagram, and the HR diagram highlighting that the intrinsic properties of stars are correlated. Now, we can estimate the distance to the stars using the distance modulus without the need for a stellar parallax measurement. The true value of this is using it for stars that are so far away that our current technology cannot measure stellar parallax.

7. Use a modified version of the distance modulus equation to estimate the distances to each of the stars. Record you calculated distances on Table 2.

𝑚−𝑀 = 5 log 789(: gives via algebra 𝑫 = 𝟏𝟎

𝒎?𝑴A𝟓𝟓 with D in parsecs

(continued next page)

To complete Table 2, you now have to use the astronomer’s version of the Stefan-Boltzmann Law, which states that 𝐿 = 𝜎𝐴𝑇* = 4𝜋𝜎𝑅C𝑇*, using the fact that stars are spherical and the surface area, A of a sphere is 4𝜋𝑅C. This looks intimidating, but it becomes much simpler if we put the luminosity, temperature, and radius in solar units. In solar units, 𝐿∗ = 𝑅C𝑇*. This is called the Luminosity-Radius-Temperature Relationship for stars. Let’s use this to finally see just how small white dwarfs are and how large these giant stars are.

8. Use the form of the luminosity-radius-temperature relationship below to calculate the radius for each star. Record these values on Table 2. For this equation, you have to use the luminosity and temperature in solar units. You get you answer in solar radii.

𝑹 =√𝑳𝑻𝟐

(continued next page)

9. Stellar parallax values have been measured for all of the stars in this lab. If the parallax, p, is measured in arcseconds, then the following equation gives the distance to the star in parsec.

𝑫 =𝟏𝒑

Use the measured parallax values to calculate the distances to the stars in parsecs. How do you distances calculated with stellar parallax (these distances) compare to those calculated with spectroscopic parallax (Table 2 Distances)

Star Parallax, p in arcsec (value ± uncertainty)

Wezen 0.00203 ± 0.00038

Iota Piscium 0.07292 ± 0.00015

Caph 0.05958 ± 0.00038

S Doradus 0.00037 ± 0.00028

Gliese 581 0.1586 ± 0.00035

Sirius B 0.37921 ± 0.00158

The parallax for S Doradus, however, was not measured until the past few years using the European Space Agency’s Gaia spacecraft, and even with the most sensitive equipment ever created, the error on the parallax measurement is nearly as large as the measurement itself. This makes for a distance determination with very high uncertainty.

(continued next page)

10. One final question before we call this lab done. Based on what you know about how spectral lines are influenced by surface gravity, and the fact that the larger the star, the lower the surface gravity (in general). Identify which spectrum in the figure is the A0 I, luminosity class I - supergiant star, and which is the A0 V, luminosity class V – main sequence star. Label each spectrum with which is which and explain your reasoning.

Which is the A0 I star and which is the A0 V star? Explain how you determined the luminosity class.