Need help with #7-#12
Calculations The slope is calculated as thus: Slope = Rise Change in y value _Ay _y2 - y1 [redshift] (1) Run Change in x value Ax X2 - X] [Mpc] The slope of our best fit line is the Hubble Constant but it is in the wrong units. To convert from [redshift/Mpc] into [(km/s)/Mpc] you must multiply by the speed of light in km/s: Slope . (3 x 105) = Hubble Constant (Ho) Mpc km/s (2) If you look closely, then, we have a unit of distance/time/distance which should equal 1/time. An inverse time? What could this strange unit signify for our non-cyclic universe? It is the inverse of the time over which the universe has been expanding. In other words: the inverse of the Hubble constant is the approximate age of the universe! We call this approximate value the Hubble Time as mentioned earlier. We can also ask ourselves an interesting question: at what distance from an observer is the universal expansion equal to the speed of light? This distance defines what we call the Hubble Limit that we discussed earlier. To calculate the Hubble time, we need to rectify the units of the Hubble Constant. The Hubble constant is in km/s/Mpc. If we change Mpc into km, we can simplify the Hubble constant into units of 1/s. Then we can convert the seconds into years. Our Hubble constant would then be in 1/years. We could flip this to say that the inverse of the Hubble Constant is in the units of years. The inverse of the Hubble Constant is the Hubble Time! Equation 3 is the simplification of this process: 1 Mpc = 3.086 x 1019km km/s 3.154 x 107 $] [km/s |year Ho Ho Mpc X 1 year = 3.154 x 10's Mpc 3.086 x 1019 [km/Mpc] 9.78 x 1011 km * year Mpc . s 978 tH billion years Ho km/s (3) Mpc To find the distance at which the Hubble limit is reached we merely need to reimagine our equation. We will determine the distance at which the expansion speed equals the known speed of light. The equation for this expansion is pretty well linear and thus we can approximate it as equation 4. From there we can develop an equation that solves for the distance, D, with a known speed, v, for the observed value of Ho (equation 5). V = Ho . D (4) c 3 x 105 [km/s] 3 x 105 Mpc 3.262 Mly 978 DH Ho [km/s] billion lightyears [km Ho 1Mpc (5) Ho Mpc Evaluations - record numerical answers with 1 decimal place. 7. Using equation 2, what is Hubble Constant Ho in km/s/Mpc? 8. If the known value is 70 km/s/Mpc what is the % error between the answer to # 7 and this? 9. If the number of data points were to increase from 20 to 200, would you say our % error would increase or decrease? 10. Using equation 3, what is Hubble Time tu in billions of years? 11. If the known value is 13.8 billion years, what is the % error of your Hubble Time? 12. Using equation 5, what is Hubble Limit DH in billions of light years
