This tutorial shows how to do the simple calculations needed to find the distance, radius, mass, etc. of stars.

DISTANCE TO STARS BY PARALLAX

1.What is parallax?

A).. B).. C).. D).. E).. |

2. The figure shows two stars observed from Earth. For which star is the
angle looking back at the Earth the smaller?

A)... B)... |

3. Which line in star A's triangle is the baseline?

A)... B)... C)... |

4. How long is the baseline?

A)... B)... C)... D)... E)... |

5. Suppose the angle of star A is cut from 1 degree to 0.1 degree but the baseline remains the same. How will distance to star A change?

A)... B)... C)... D)... |

6. Suppose the angle is cut by a factor of 50. By what factor is the distance changed?

A)... B)... C)... |

7. If the parallax angle is smaller, is the star farther or nearer?

A)... B)... |

FIGURING OUT THE SIZE OF A STAR FROM ITS BRIGHTNESS

The amount of light a star emits depends on its surface area. Other things being equal, a star with a big surface area will emit more light than a star with a small area.

Assuming that a star is a sphere, its area is set by the **square** of its
radius. That is, a star whose radius is twice that of another star has four (2x2) times the surface area.

The questions below are designed to give you a feeling for how astronomers figure out relative sizes of stars from their brightness.

For simplicity, let's start with flat glowing objects rather than spherical stars.

8. A hot piece of glowing metal is 1 meter square and gives off 100 watts of energy. How much energy is given off by a square piece 5 meters by 5 meters?

A)... B)... C)... D)... E)... |

9. Suppose now you measure the amount of energy given off by a piece of hot glowing metal identical in temperature to the one above and discover that it is radiating 2000 watts. How many square meters is it?

A)... B)... C)... D)... E)... |

Notice that the amount of energy given off depends on the area that is radiating.

Notice also that we can determine the area from the amount of radiated energy, other things being equal.

The next few questions should help you see the relation between *area* and size. That is, between area and radius for a circle or area and length of an edge for a square.

10.What is the area of a square 2 meters on a side? (See figure.)

A)... B)... C)... D)... E)... |

11. What is the area of a square 5 meters on a side?

A)... B)... C)... D)... E)... |

12. What is the dimension along one edge of a square whose area is 36 square meters?

A)... B)... C)... D)... E)... |

13. What is the dimension along one edge of a square whose area is 144 square meters?

A)... B)... C)... D)... E)... |

Notice that the area (A) = the edge length (L)squared. That is, ** A = L^{2}**.

This is an example of a scaling law. That is, once you know that area goes as the square of the dimension, you can quickly calculate that doubling the dimension raises the area by 2 squared, or a factor of 4. Similarly, making the dimension 5 times bigger, makes the area 5 squared or 25 times bigger.

Now lets apply this to stars. The Sun gives off 4x10

14. If a star gives off 4 solar luminosities of energy and is like the Sun in all ways except its surface area, what must its surface area be compared to the Sun's?

A)... B)... C)... D)... E)... |

Stars are, of course, not flat squares. They are approximately spheres. However, the same general scaling laws work for spheres as well as for squares. That is one reason scaling laws are so powerful.

For a sphere, the area scales as the square of the radius. In particular,

the surface area of a sphere = 4R^{2}. Notice this formula, like that for the square has area given by a quantity involving a length squared. **Areas always depend on the square of a length. **

15. Find the area of a sphere whose radius is 1 meter.

A)... B)... C)... D)... E)... |

16. Find the area of a sphere whose radius is 3 meters.

A)... B)... C)... D)... E)... |

17. Divide the area of the 3 meter sphere by the area of the 1 meter sphere. What is the ratio of their areas?

A)... B)... C)... D)... E)... |

18. Would you have gotten this answer by simply multiplying the first answer by 3 squared?

A)... B)... |

19.Given that the star above has 4 times the Sun's surface area, how many times bigger must the Star's radius be than the Sun's?

A)... B)... C)... D)... E)... |

20. Suppose a star has 100 times the Sun's surface area. How many times bigger must the Star's radius be than the Sun's?

A)... B)... C)... D)... E)... |

Such calculations are how astronomers find the radius of a star.

TAKING A STAR"S TEMPERATURE

Another property of a star that can be measured easily is its temperature. We have already discussed measuring temperature of hot objects when we were talking about light. We saw then that an object's temperature can often be deduced from its color using Wien's Law.

The basic idea is that hotter objects radiate more strongly at shorter (bluer) wavelengths. Thus, other things being equal a bluish star is hotter than a reddish star.

21.Star A has a surface temperature of 15,000 K while star B has a surface temperature of 4,500 K. Which star is the red one?

A)... B)... |

To determine the star's temperature from its color, astronomers measure the brightness of the star at a series of wavelengths and then determine at which wavelength it emits the most light; that is, at which wavelength is it brightest. We will call that wavelength _{m}, where the m stands for "maximum."

To get the temperature, we use Wien's Law. T = 3x10^{6}/ _{m}, where T is in Kelvins and _{m} is in nanometers, abbreviated nm.

For example, suppose we want to measure the temperature of the bright star Rigel. Measurement of its light shows that _{m} = 250 nm. What is its surface temperature?

Plugging this value of _{m} = 250 nm into Wien's law gives

T = 3x10^{6}/ _{m}

T = 3x10^{6}/250 = 12,000 K.

22. Given this temperature for Rigel do you expect it to be bluish or reddish?

A)... B)... |

23. For another example, suppose we measure the brightness of Betelgeuse and find that it radiates most strongly at about 1000 nm. What is its temperature?

A)... B)... C)... D)... E)... |

MEASURING A STAR"S MASS

Another property of a star that astronomers often need to know is its mass. Astronomers can find a star's mass relatively straightforwardly if the star has a companion star in orbit around it. The method is based on the fact that the **gravitational interaction between the stars depends on their masses and that the gravitational interaction also determines their orbital properties.**

Specifically, if two stars orbit one another, a modified form of Kepler's third law relates their masses, separation, and orbital period (the time to complete an orbit).

The law states that M+m = P^{2}/a^{3}, where M and m are the two masses in solar units, P is the orbital period in years, and a is the
semimajor axis of their orbit in astronomical units (AU).

Note, that if we have just a and P of the orbit, we cannot find the individual star masses, only their combined mass. The individual masses can be found if we know additional things about the orbit, which we will ignore here.

Example: Suppose two stars have an orbital period of 10 years and their orbit has a semi-major axis of 2 AU. What is their combined mass?

If we put these into the modified form of Kepler's third law we get M+m = P^{2}/a^{3} = 10^{2}/2^{3}=100/8=12.5 Solar masses.

24.What is the combined mass of two stars orbiting with a period of 25 years and having an orbital semi-major axis of 5 AU?

A)... B)... C)... D)... E)... |

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