(e) (3 points) For which equation were your calculated k values about the same for both stars. This tells us that this is the appropriate relationship between L and R. Take the average of your two k values for this equation and plug it into the appropriate original equation listed in part 2d; be sure to leave L and R as variables (i.e. letters). Text Stellar Luminosity and Temperature (24 points total): 3. Now let's investigate how a star's luminosity and surface temperature depend on one another while keeping its radius constant. (a) On the interactive H-R diagram, return your star marker to its original position (Star A's values). Copy your data from the first table into the table below. (b) (3 points) Click and drag your star marker down its green line of constant radius to a location that has the same radius and half the temperature of Star A. Double-check that that the radius has remained the same. In the table below, record your new star's luminosity, radius, and temperature in the row labeled \"Star D.\" Stellar Choice | Luminosity (Lo) | Radius (Ro) Surface Temperature (K) Star A Star D () (3 points) Has halving the temperature of Star A increased or decreased the luminosity? By what factor has the luminosity increased /decreased? (d) The following equation predicts the relationship between a star's radius (7' measured in units of Kelvins) and its luminosity (L measured in units of solar luminosities, Lo) where k is a proportionality constant: L=kT* (1) i. (4 points) Plug Star A's luminosity and surface temperature into this equation to solve for the associated constant k. Be sure to show your work and include at least 4 non-zero digits in your answer. ii. (4 points) Similarly, plug Star D's luminosity and surface temperature into this equa- tion and solve for its associated constant k. Be sure to show your work and include at least 4 non-zero digits in your answer. () (5 points) Take the average of your two k values for this equation and plug it into the original equation from part 3d; be sure to leave L and T as variables (i.e. letters). Use this constructed equation to predict the luminosity of a star (let's call it Star E) that has the same radius as Star D but a temperature that is one-fourth of Star D's surface temperature. Show your work and your answers in the appropriate cells below. Stellar Choice | Luminosity (L) | Radius (Ro) Surface Temperature (K) Star E - Predicted Star E Measured (f) (2 points) Click and drag your star marker down its green line of constant radius to a location that has the same radius and one-quarter the temperature of Star D. Double-check that that the radius has remained the same. In the table above, record the star marker's luminosity, radius, and temperature in the row labeled \"Star E ~ Measured.\" (8) (3 points) How do your predicted and measured Star E luminosities compare? Where do you think any discrepancies came from? Stellar Types on the H-R Diagram (8 points total): 4. On the interactive H-R diagram, manipulate the temperature (bottom) and luminosity (left) sliders and note the corresponding changes in the position and appearance of the star marker. In the table below, check the appropriate region(s) of the H-R diagram that correspond(s) to cach description. Hint: Four rows will have two bozes checked. Stellar Description | Upper Left | Upper Right | Lower Right | Lower Left Cool Stars Hot Stars Faint Stars Luminous Stars Small Red Stars Really Large Red Stars Small Blue Stars Large Blue Stars The Main Sequence (9 points total): 5. The thick green curve that passes through the interactive H-R diagram is known as the main sequence and contains all stars that are fusing hydrogen into helium as their primary energy source. Over 90% of all stars fall in this region on the H-R diagram. Move the star marker up and down the main sequence and explore the different values of stellar luminosities, tempera- tures, radii, and color. Circle the options below to describe the stars along the main sequence in terms of these characteristics. (a) What are stars like near the TOP of the main sequence? Radii: small (0.1-1Ro) medium (~1Ro) large (1 - 10 Rg) Luminosities: low (0.01 - 0.001 Lo) medium (1 - 10 Lo) high (1,000+ L) Surface Temps: low ( 10,000 K (b) Colors: red orange yellow yellow-white white blue-white blue (c) Spectral Types: M K G F A B O (d) Luminosities: ~ 1,000Le () Radii: 0.1-1Rg 100R; 100 1,000R; > 1,000Re 7. (5 points) Use your answers from all parts of question 6 to generalize the properties of our closest stellar neighbors. Are they hot or cool? Are they big or small? Are they bright or dim? Are their properties similar to those of our Sun