Astronomy Homework

Lab 1 – The H-R Diagram

The Hertzpsrung-Russell (H-R) diagram is a way of categorizing properties of all stars in the sky, including the Sun. The surface temperature, color, luminosity, and radius of stars can all be deduced from their location on the H-R diagram. When a star begins to condense from an interstellar dust cloud, it enters the HR diagram in the middle right. This protostar collapses toward the Main Sequence, taking tens or hundreds of millions years, until the core becomes hot enough to support hydrogen fusion. Collapse stops, hydrostatic equilibrium is attained, and the star has now become a Main Sequence star, which burns hydrogen into Helium in its core. Hot, massive O-type stars only spend a few million years on the Main Sequence; a cooler G-type star like the Sun will live on the Main Sequence for about 10 billion years; low-mass, cool M stars can survive as Main Sequence stars for 100 billion years or longer. When a star has used up most of the hydrogen in its core, it swells up and becomes a giant. The most massive stars produce supernovas; Sun-like stars die more gently and their cores become White Dwarfs.

Lab Tools

We will use the Hertzsprung-Russell Diagram Explorer developed by the University of Nebraska-Lincoln for this lab. This explorer shows a Size Comparison in the left panel between the properties of a selected star and the sun. In the panel you can adjust the temperature and luminosity of the other star and it will give you that star’s radius in solar units. The right panel shows the HR Diagram, you can adjust what the axis are under Options, but it is fine to stay with temperature versus luminosity.  You can also click on and off other quantities shown on the plot.  Under Plotted Stars you can select which stars are plotted on the diagram.  The red x on the diagram starts at the position of the sun, you can move it by dragging it or setting its properties in the Size Comparison panel.

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Assignment

First click on the brightest stars under Plotted Stars. Then click on the nearest. Why do these two choices occupy such different places in the HR diagram?  Drag the red x to the coldest part of the main sequence. What is the size of a star with this temperature and luminosity. Drag the red x to the hottest value on the main sequence. What is the size of the star now?

Click on show luminosity classes. What is the range of temperatures and sizes for supergiants? How about for red giants?  Click on show instability strip, this will show the region where pulsating variables live. What is the largest and smallest size stars in this region.

Questions

  1. If a star is the same temperature as the sun, but 10000 times more luminous what is its size?
  2. If a star is 10000K and one thousandths the sun’s luminosity, what is its size?
  3. How many times more luminous is the most massive star than the least massive star? How many times hotter is it?

 

 

 

 

Lab 2 – Eclipsing Binaries

Eclipsing binaries, binary stars where the orbit causes one to pass in front of the other from our view, are the main way astronomers measure the sizes of stars.  When a star passes in front of another star it will dim a little just like when a planet transits a star.  The amount of time it takes for the brightness to reach its minimum and the amount of time it stays at the minimum tell us about the sizes of the two stars.

Lab Tools

For this lab we will use the Eclipsing Binary Simulator developed at the University of Nebraska – Lincoln.  In the upper left box of the simulator we see the perspective from Earth.  The upper right box shows the light curve as a function of phase which is just the fraction of the stars orbit.  This is normalized to be 1.0 when there is no eclipsing.  The bottom left panel allows you to control the System Orientation and the Animation and Visualization Controls.  Here you can set the longitude and inclination of the binaries orbit. Note that at most inclinations the binary will not be an eclipsing binary.  You can also start and stop the animation and control its speed.  You can click off the lock on perspective from Earth, which will let you drag the orbit so you can see it from a top down view without effecting the light curve.  The right bottom panel controls the properties of the stars and their orbit. There are a number of preset choices and you can control the mass, size and temperature of each star as well as the separation between the stars and the eccentricity of their orbit.

Assignment

Let us start with preset Example 1.  This is the case of two identical stars. What feature of the light curve can be used to measure the size of the star?  Now switch to preset Example 2. In this case what features of the light curve can be used to measure the size of the smaller star? How about the bigger star? Move on the preset Example 3. What is different about this light curve? What star property causes the difference? Now let’s try preset Example 6, notice we skipped Example 4 and 5. What is different about the light curve now? What is causing the difference,  you can change the inclination of the orbit to help you see what it is? Finally let’s look at Example 7. Change the first stars temperature to 5000K and the orbit eccentricity to 0.3. Now look at the light curve. Can you explain why it looks like this? This is actual a more realistic example then the previous ones which are simplified to make it easier to see what is going on.

 

Questions

  1. In preset Example 1 what is the minimum inclination where you can still see this configuration as an eclipsing binary?
  2. What effect does changing the star’s mass have on the orbit?
  3. What effect does changing the separation have on the light curve?

 

 

 

Lab 3- Exoplanets

There are two main methods for discovering exoplanets (planets around other stars). The first is the radial velocity technique which detects the a wobble in the stars radial velocity because of the gravitational effect of the planet. The second is the transit technique which detects the faint dimming of the star as the planet passes in front of it. Both techniques are challenging because they require the correct alignment and the signal is small, so very precise instruments are needed.

Laboratory Tools

In this lab we will be making use of two simulators created by the astronomy department at the University of Nebraska-Lincoln.  Note that in both these simulators the scale on the y-axis changes so pay attention to it. The simulators are:

  • TheExoplanet Radial Velocity Simulator – this simulator show the radial velocity technique for identifying exoplanets. The upper left box shows the planet in orbit and the direction from which we are viewing it. The upper right panel shows a plot of the radial velocity of the star versus the phase which is just the fraction of the orbit the planet has gone through. You can check boxes to show the theoretical curve and to show simulated measurements.  The amount of noise and number of data points can be controlled if the box for show simulated measurements is checked. On the bottom left there is a box Animation Controls that lets you start and stop the animation which just makes the planet go around.  In the bottom middle you have boxes for System Orientation and Star Properties. These controls can change the angle the system is being viewed from and the mass of the star.  The bottom right has a panel Presets that has drop down menu of preset properties for the planet and a panel Planet Properties to set those properties yourself.
  • TheExoplanet Transit Simulator – this simulator shows the transit technique for discovering exoplanets. The upper left box shows a picture of the star and the planet transiting it. The upper right box shows a plot of the normalized flux from the star, with 1.0 being the flux when there is no planet transiting.  You can check boxes to show the theoretical curveand to show simulated measurements.  The amount of noise and number of data points can be controlled if the box for show simulated measurements is checked.  In the middle on the left side is a box, Presets, which lets you choose preset values for the planet’s properties from a dropdown menu. Below it are the Planet Properties which you set individually.  In the middle on the bottom is a box for Star Properties which lets you set the star’s mass and in the bottom right there is a box System Orientation and Phase which lets you set the inclination and longitude of the system.

Assignment

Let us start with the Exoplanet Radial Velocity Simulator.  Start with only showing the theoretical curve which is easier to follow and with preset A that has the star’s mass as 1.0 solar mass, the planet’s mass as 1.0 Jupiter mass, the semimajor axis as 1.0 AU and an eccentricity 0.0.  Record the maximum and minimum radial velocity on the plot. Now change the mass of the star from 0.2 to 2.0 solar masses. How does this effect the radial velocity?  Return the star’s mass to 1.0 and now change the planet’s mass, what happens?  Return the planet’s mass to 1.0.  Adjust the semimajor axis from 0.01 to 10.0, what is the maximum radial velocities. What is the maximum radial velocities in these cases? Return the semimajor axis to 1.0 and change the eccentricity to 1.0, What happens to the radial velocity plot?  Return the eccentricity to 0.0 and change the inclination between 0 and 180. How does the inclination effect the observation? Change the longitude, this should have no effect when the eccentricity is 0.0.  Change the eccentricity to 1.0 and now adjust the longitude again. What effect does this have on the radial velocity?

Now let’s switch to the Exoplanet Transit Simulator. Again start with only showing the theoretical curve which is easier to follow and with preset A that has the star’s mass as 1.0 solar mass, the planet’s mass as 1.0 Jupiter mass, the semimajor axis as 1.0 AU and an eccentricity 0.0.  What is the lowest value for the normalized flux?   How long does the eclipse (transit) last? Now change the planet’s mass, planet’s radius, semimajor axis and the eccentricty. How does each effect the flux and length of the eclipse? Reset to the original values. Now change the star’s mass from 0.5 to 2.0 solar masses. What effect does this have on the flux and length of the eclipse? Finally adjust the inclination, don’t use the slider you need to make small changes.  Change the inclination to 90.1, 90.2, 90.25 and 90.3. What effect does inclination have on observing the planet’s transit?

Questions

  1. What combination of properties leads to the easiest detections of extra solar planets using the radial velocity technique?
  2. What combination of properties leads to the easiest detections of extra solar planets using the transit technique?
  3. For the transit technique what disadvantage does a more massive star cause for detecting an exoplanet? What advantage do you gain?

 

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