ASTR1020 -- HOMEWORK 5
DUE MONDAY APR. 6th

WEIGHING GALAXIES

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Note: Feel free to write your answers on a separate sheet of paper as you work off of the web page on a computer. If you want to print out the assignment anyway, adjust your browser so that it is about the same width as the above line. Several people have had difficulty printing out the assignment, this is in part due to browsers' poor handling of frames. Click here if you want to open the homework in a frameless window. I have put the figures at the end of the assignment to ease the problem further. I strongly recommend that you check the homework you print against the web page to make sure everything printed! You can get help with this assignment during office hours (note that David's have changed) We also will have a help session on Wednesday evening, Apr 1st, 5:30 - 8:00 PM at SBO downstairs.

Introduction:

As we discussed in class, astronomers can infer the masses of astronomical systems by measuring the velocities of objects that are orbiting the system in question. Therefore, for example, one can infer the mass of the Earth from the orbit of the Moon, and the mass of the Sun from the orbit of the Earth. The basic principle is described in your text. If an object is in a circular orbit, so that the gravitational attraction is equal to the centrifugal "force", then the mass of the central attracting object is given by:

(A)        M=V²R/G

where V is the orbital velocity, R is the radius of the orbit, and G is Newton's constant of gravity. (This equation follows directly from the equations in section 2.7 of your text.) If distances are measured in meters (m) and velocities in m/second, and G has the value given in Appendix A3 of your text, then the resulting mass of the attracting system will be measured in kilograms.

1) Test the validity of equation (A) by calculating the mass of the Sun from the Earth's orbit. The velocity of the Earth's orbit is 29.9 km/s. Convert that number to m/s by multiplying by 10³ (the number of meters in 1 kilometer). Likewise, look up the radius of the Earth's orbit¹ (1 Astronomical Unit) in lesson 3 in km and convert to meters by again multiplying by 10³. Then plug these values into equation (A) to calculate the Sun's mass (M_sun). Show your work and result below. Compare your answer to the value given in lesson 3. Be careful of the units, namely, make sure you are using meters and seconds and kilograms in your calculations. You don't have to worry about the "N" in Newton's Gravitational constant, though if you want to, this "N" is short for (kg m)/s².






2) Now we are going to weigh a galaxy! Equation (A) can be used to find all the mass inside a given orbit. In the case of the Earth's orbit around the sun, this is pretty much the mass of the sun only. If we apply it to matter (gas usually) orbiting the center of a galaxy, we measure all the mass interior to that gas. By looking at the rotation curve (section 23.8 in your book) of a galaxy we can figure out what the total interior mass is as a function of radius. You are going to do just such a series of calculations for the galaxy NGC 247².

FIGURE 1 TOP A visible image of NGC 247; a spiral galaxy in the nearby group, Sculptor. Image courtesy of the Digitized Sky Survey (DSS). BOTTOM This "false" color image of NGC 247 was created from the DSS image in the blue channel and an HII(red) and OIII(green) image from Mike Dopita's (Australian National University) pretty picture page. The foreground stars have green halos due to relatively poor focus in that channel. The outermost red blobs (ionized hydrogen regions) are about 8 kilo parsecs from the center of the galaxy.

Below in figure 2 is the rotation curve of the NGC 247. At the eight points highlighted in red and marked with vertical hash marks you will calculate the mass of the galaxy interior to these points but first answer this question:

The position of the last filled in solid black circle corresponds to where the glow of the galaxy appears to vanish. If you didn't have those points at larger radii to look at would you expect the points further out to have lower or higher velocities? Why? (Hint: Think about how equation (A) changes as r increases)

Now use equation (A) to calculate the masses. Convert these masses to solar masses (i.e.  the number of suns would it take to equal the same mass) using 1 M_sun =2E30kg (which should be what you got in part 1). Because the units on the graph are kpc and km/s instead of meters and m/s you must use this modified version of the "G" constant converted to these units; G=2.16E-36!!! This will give you the answer in kg. ("E" is short for "times ten to the"). You may wish to use a spreadsheet program³ if you think this will be easier than calculating on a calculator. If you do use a spreadsheet program, you may wish to plot a graph of the eight points as you have them entered in the spreadsheet to verify that you did not make any mistakes in your estimates or typographical errors. Fill in your results in the table below. Remember you are calculating the mass of an entire galaxy by the time you get out to 3-4 kpc, so check and see if your answers are reasonable!

FIGURE 2 This is the rotation curve of NGC 247. The vertical axis is velocity of rotation and the horizontal axis is distance from the center in kilo parsecs (1kpc=1000 parsecs 3.086E21 cm). The solid line represents what the rotation curve might look like if the light (visible and radio) from the galaxy was a good indicator of how much matter was actually there. The different symbols represent data gathered in different ways. The filled in circles are from spectral observations of ionized hydrogen gas, the empty circles are radio telescope observations of neutral hydrogen gas; for now this is unimportant.

Now answer the following questions:

• Do you think the mass you came up with is larger than, smaller than or about the same as the mass represented by the solid line in figure 2? In other words, if you converted data from the solid line into mass estimates how does it differ from what you calculated from the points. Explain your reasoning.
  
  
  
  
  
• If there is a significant discrepancy betwen the velocity that was measured (the points) and what we think the velocity should be based on the matter we see (the solid line) what might explain it? If there isn't a significant discrepancy what does that mean?
  
  
  
  
  
• Extra Credit: Around 4 kpc there is a large discrepancy between the neutral hydrogen (HI) data and the ionized hydrogen (HII) data. What might explain this? (HINT: Think about what how large HII regions form.)
  
  
  
  
  
  
  

3) At the center of NGC 4258, something peculiar is going on. From measurements of the velocity of gas in a circular orbit around the around the center, you can calculate the mass in the very center of the galaxy. These figures are a recent measurement of the velocity around the center of the galaxy NGC 4258, inferred from the Doppler shift of radio spectral lines emitted by water vapor molecules in the gas. In figures 3 and 4, an angular distance of 1 'mas' (micro-arcsecond) corresponds to a radius R = 9.5 x 10¹⁴ meters. Note that the center of the galaxy is moving away from us with a LSR (velocity relative to the Sun's "Local Standard of Rest") velocity of 475 km/s, and you should subtract this central velocity from the LSR velocity to measure the circular velocity from the graph. Now, read the radius of the rotating points of gas from the graph (in mas; remember that you care about the absolute distance from the center, so minus signs should be ignored), convert the values to meters (m), and calculate the mass of the central object (in kg) the same way as you calculated the Sun's mass. Then divide this number by M_sun (the mass of the sun that you found in part 1) to find the mass in units of M_sun (how many times the sun it weighs). Show your work below for one of the five data points and then fill out the table below with the entire set of the five highlighted in red (and marked with horizontal hash marks) points.

Now calculate the average of all five masses you put in the table in kg, then convert to M_sun units.









Extra Credit: In class we discussed the Schwarzschild radii of Black Holes. How does the maximum possible radius of the central object in NGC 4258 compare to the theoretical radius of a black hole with the mass you calculated above? Show your work! Many scientists think a black hole is the only explanation for the central mass of NGC 4258. Why are they so sure?

FIGURE 3 (Top) A schematic of the center of the galaxy NGC 4258 as it might look from "above" the disk. The black dots are positions of the emitting water vapor. In both figures 3 and 4, the points highlighted with red are the ones you will use in the assignment.

FIGURE 4 (Bottom) Figure 4: A velocity-position plot (the velocity of water vapor plotted against its position on the sky) of the emitting water vapor. Use the vertical and horizontal lines to help you estimate the position of the emitting gas in velocity and distance from the center. Don't forget to account for the LSR velocity of 475 km/s. of the whole system! (Figures from Miyoshi, M. Moran, J., Herrnstein, J., Greenhill, L., Nakai, N., Diamond, P., Inoue, M.,Nature, 373, 127, 1995)

²NGC stands for "New General Catalogue". this catalogue was created around the turn of the 20th century, I think. The 247 simply means that this galaxy is the 247th entry in the catalogue

³If you want to see the hints on using Excel back in homework 3, here are links to tips on entering formulas and making charts in Excel.