Reverberation Mapping Astro 7B Spring 2016, Lab 101 GSI: Goni Halevi SOLUTIONS 1 What is it? As a quick review, let s recall that the purpose of reverberation mapping is to measure black hole masses. In order to use reverberation mapping, you need to know that the black hole is accreting, so it s used often for AGN (active galactic nuclei). The basic equation for the mass of an object is M = v2 R G where v is the velocity of a circular orbit at radius R. Reverberation mapping is particularly cool because it s independent of distance, and measuring distances is tough. 2 Recalling Equations Note: try to do this part without looking at your notes. a) Look at Equation 1. To find mass, we need to measure radius (R) and velocity (v). In reverberation mapping, we measure R. How? What is the formula to find for R? Find the time lag between the continuum and a spectral line of choice. The radius of the broad line region (BLR) for the spectral line you look at is given by R = c t where t is the average time lag measured. b) What method do we use to find the velocity v? What formula do we obtain? Be sure to explain what the variables in your formula represent. Doppler shift! We need to use the same spectral line as we did to find radius, let s say it has rest wavelength (as measured in labs) denoted by λ 0. Then, the line we actually see in the spectrum will have some broadness that comes from the fact that the region is rotating around the black hole, so different parts are moving at different velocities along our line of site. This causes the line to broaden as it is made up of components with different velocities and thus slightly different wavelengths. We typically measure the thickness of this spectral line as the full width half-maximum (FWHM), which just means we take the full width of the line at a point halfway to the peak. Denoting this broadness as λ, we have a velocity v given by v c = λ v = λ c λ 0 λ 0 (1) c) What are the limitations of this method for finding velocities? (Hint: consider inclinations). Doppler shift allows us to measure velocities along our line of sight, so we measure the true rotational velocity if the disk is edge on, but can t measure it at all if it is face on, and measure less than the true velocities for all other possible inclinations since we can only measure one component. 1
d) Plug in your formulas from a) and b) to rewrite Equation 1. M = v2 R G = ( λc λ 0 M = λ c 3 t λ 0 G ) 2 c t G e) Use dimensional analysis to make sure that your equation in part d) makes sense. The ratio of broadness to rest wavelength is dimensionless. c 3 has dimensions of cubic length over cubic time, while t has units of time, so we have cubic length over time squared in the numerator. The units of G, if you don t have them memorized, can be found by considering the equation for Newtonian gravitational force. F g = GMm R 2, so the units of G must be the units of force times length squared divided by mass squared. Force is mass times acceleration, so mass times length divided by time squared. Multiplying by length squared and dividing by mass squared, we find that G has units of inverse mass times length cubed times time squared. Overall, we have cubic length and time squared in the numerator, and then inverse mass, cubic length, and time squared in the denominator. Cancelling like dimensions, we are left with mass, as desired. 3 Real Reverberation Mapping: Time Lag You ve derived the formula for determining mass. However, in practice, things are hardly that simple. We re going to examine the case of a particular active galactic nucleus, the one in a galaxy called Mrk 40. To measure time lags, we gather data on the AGN over time and compare spectral line flux to that of the continuum, looking for delays we can measure. From the way we ve talked about the method in class, it seems like it d be straightforward to measure time lags, but it isn t really. To illustrate this, I ve included Figure 1, which I ll leave to your interpretation. a) Interpret Figure 1 using your knowledge of reverberation mapping. What does each subplot represent? What are the axes? The x-axis is time (NOT WAVELENGTH) in days, with a constant offset. The y-axes are flux measurements. Each subplot represents a different spectral line (the one labeled in the top left of the subplot), except for the top-most subplot. That one shows the continuum in UV wavelengths. From the data presented in Figure 1, you can determine the radius of the broad line region (BLR), but it s not such a simple task. To determine time lags, we do something called crosscorrelation. Mathematically, the cross-correlation of two functions f(t) and g(t) is f g(τ) + f (t)g(t + τ)dt where f (t) is the complex conjugate of f(t) and τ is the lag. 2
Figure 1: Some data for the reverberation mapping of Mrk 40. b) How would cross-correlation techniques simplify the calculation of lag times? Explain. You find out how much the two light curves overlap as you shift them. If they are the same, except for a time lag the overlap is maximized for a lag equal to the time lag. Note that because you re dealing with real data, your cross-correlation function is not maximized at 1 (which would indicate total overlap), but near 1. Do look up cross-correlation and read more about it if you d like. Also feel free to ask me about it. c) Given that Figure 2 shows the cross-correlation functions for different spectral lines as compared to the V band magnitude (shown in black), approximate the time lag for each line and sketch an accretion disk labeled by element that would create this array of time lags. This is just a matter of looking at Figure 2 and determining the approximate locations of the peaks for each curve (except for the V curve, which is the continuum and peaks at 1 because it is being cross-correlated with itself). Here are my estimates (don t worry if yours aren t identical): 3
Hα: 9 10 days Hβ: 5 days Hγ: 4 days Hδ: 3 days HeII: 0 days F eii: 10 11 days d) Which line would be the best to use for reverberation mapping of Mrk 40? Why? The Hβ line is by far the best choice, for a number of reasons. For example, Its peak is clearly defined in the cross-correlation function. It has a discernible time lag, not too short. Its overall flux is much more than Hγ s or Hδ s (see Figure 1). It peaks at the highest cross-correlation, indicating that it resembles the continuum more closely than the rest of the spectral lines do. Figure 2: Cross-correlation functions for Mrk 40. 4 Thinking about Limitations... Figure 3 shows a correlation between the size of the BLR for Hβ and the luminosity at a specific wavelength of 5100 Å. 4
Figure 3: There is a correlation between BLR size and luminosity of AGNs. Note that this is a log-log plot and that the x-axis has units of ergs/second. a) Determine the Hβ BLR size in light days for Mrk 40. Don t over-think this one. I approximated the time lag for Hβ as t = 5 days, so the radius of the Hβ broad line region is R = c t = 5 light days. b) What value of λl λ (5100 Å) does this BLR size correspond to, approximately? Going up to 5 light days on the y-axis and across to the black least-squares fit line, then down to the x-axis, I find λl λ (5100 Å) 3 10 42 ergs/second. c) If you wanted to use reverberation mapping for an AGN with λl λ (5100 Å) two orders of magnitude greater than that of Mrk 40, what is the minimum time you would need to collect data for? Two orders of magnitude more luminous 3 10 44 ergs/second. Going to this value on the y-axis, up to the black line, and across to the x-axis, I approximate a BLR radius of 70 days. d) What does this tell you about what kind of AGN are easier to find BH masses for? Less luminous AGN Need to watch for a shorter time period Easier to get your telescope time proposal approved Easier to obtain data Easier to calculate BH masses for. 5
5 Real Reverberation Mapping: Mass Measurement a) Suppose you measure the Hβ line of Mrk 40 to have a FWHM of 0.06 nanometers, given that the rest wavelength of Hβ is 486.10 nanometers. What mass do you determine for the BH in Mrk 40, assuming it is edge on? (Answer in solar masses). I worded this badly before. We don t care about the absolute location of the spectral line, because that s caused by cosmic redshift distant galaxies move away from us, so their spectra are shifted to redder wavelengths. Instead, we care about the spread of the Hβ line, representing the various velocities contributing to that line and thus the rotation of the accretion disk. Thus, λ = 0.06nm, λ 0 = 486.10nm M = λ c 3 t λ 0 G = 0.06 c 3 (5 days) 486.1 G M = 2.15 10 37 kg 10 7 M. b) If instead, we have no idea what the inclination of the accretion disk is, what is the range of possible masses for the BH in Mrk 40? We measure only the line-of-sight component of the velocity. Let s call the true velocity v and the one we measure v. Then, we see v = v sin i where i is the inclination (π/2 for edge-on, zero for face-on). This can be determined through simple trigonometry, draw some triangles if you don t know where I got this. Inverting, the true velocity is related to the measured velocity by v = v sin i. This is consistent because when the disk is edge-on, we get v = v and the velocity we calculate is the true velocity. In the case where the disk is face-on, sin i is zero so we can t use this equation, because we can t measure any velocities. Since sin i can be less than 1, but never more, v < v for all i. This tells us that the velocity we use, v, is always less than the true velocity. Since the formula for mass depends on this velocity as M v 2, we can only get masses less than the true mass if we don t know the inclination. Thus, the mass we calculated is a lower-limit on the possible mass of the black hole. c) In the next LAMP campaign, we want to measure black holes with masses that are an order of magnitude greater than that in Mrk 40. How does this affect the telescope time proposal? Well, more massive black holes have more luminous accretion disks. This is intuitive since assuming the same efficiency, more mass means more light to make. It can also be shown mathematically since we ve investigated in part 4 of this worksheet that the BLR radius grows with luminosity, and we know M R, so mass must also be positively correlated with luminosity. Like we said before, more luminous AGN need to be watched for more time because their time lags are longer since the BLR radius is greater. Thus, we needed to ask for much more telescope time this time around. 6
Congratulations on making it to the end of this quite long worksheet! If you re particularly interested in reverberation mapping, talk to me about it and I can show you the LAMP proposal and elaborate on it even more than this worksheet does. I may be biased, but time domain astrophysics is the coolest astrophysics and reverberation mapping is just one example of a much larger sub-field that s very actively researching right now. Also, if you liked the cross-correlation technique presented in this worksheet, it s just one example of how computational programming can simplify data analysis dramatically, and if you re an intended astro major planning to take optical lab, you ll get to use it cross-correlation for a totally different application. 7