The best way to estimate H_0 is from the best-line fit to the data. Use Excel to
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The best way to estimate H_0 is from the best-line fit to the data. Use Excel to plot the points. Put distance d (Mpc) along the x-axis and recessional velocity v_r (km/s) along the y-axis. If your unfamiliar with how to do this, here's a tutorial: Excel Tutorial for 2000 (HTML), or PDF for 2007. In an ideal world, the data would all lie on a straight line. The slope of that line through the data points is the Hubble Constant H_0 in km/sec middot Mpc. However, as often happens, the points from our experiment do not lie along a straight line. In these type of situations, we find the best straight line to fit the data and calculate its slope. The slope of a straight line is the ratio slope = vertical increment/horizontal increment. Use Excel to find the best-fit straight line, or trendline, for your data. From the equation of the line that Excel finds for you, figure out the slope of the line. This is your estimate for H_0. There are instructions on how to do this in the Excel Tutorial HTML for 2000 or PDF for 2007. Part A - The Hubble Constant from the best-line fit Using a spreadsheet program, find the best-fit straight line to you data. From the slope of your best-fit line, what is the value of the Hubble constant H_0 for your data? H_0 = _______ km/s/Mpc Part B-Show your best-fit line Using the equation for your best-fit line, find the predicted recessional velocity for galaxies at these distances: 10, 50. 100, 200, and 300 Mpc. Plot the points based on your best-line fit. The points do not need to be plotted in the exact correct location but put them as close as possible to the calculated value.Explanation / Answer
Solution to find the best fit line and the slope (Hubble's constant):
A. Start Microsoft Excel.
B. Enter data into Excel spreadsheet as the first column (A) containing values of x, i.e. Mpc whereas the second column (B) containing values of y, i.e. recessional velocity in km/s
C. Highlight all cells containing data.
D. From the "Insert" tab select "Charts - Scatter". Use the first type of scatter charts – “Scatter with only Markers”. A simple plot will be prepared by Excel.
E. Next step is to add axis labels and legend to the graph. Select “Layout” tab from “Chart Tools”. Then add a header using the “Chart Title” button and add axis labels using “Axis Titles” button (both for horizontal and for vertical axes). Here, horizontal axis will be Distance (Mpc) and vertical axis will be Recessional Velocity (km/s).
F. The last step is to add the linear fit (a straight line fit) to the graph (chart). Click once anywhere inside the graph area. Select the “Layout” tab from “Chart Tools”. Click on the “Trendline” icon and select the “Linear Trendline” option. A graph appears.
To show the equation, click on “Trendline” and select “More Trendline Options…” Then check the “Display Equation on chart” box.
From the equation for that straight line (y = mx + c) we can conclude that the best estimate of the Hubble's constant is: H0 = m (km/sec.Mpc), where H0 is the slope of the line and c represents the y-intercept.
This gives us the value of H0. The values of recessional velocities for distances can be calculated with the line's equation Velocity 'y' = H0x + c, where c is the intercept found above and x would be 10, 50, 100, 200, 300 Mpc.
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