## Building and Applying Mathematical Models of Angular Size

**Overview:** Students learn to use the angular size relation to predict linear sizes of terrestrial objects before applying the method to astronomical objects.

**Electronic resources:** FITS images of common objects, angular_sizes.fits (FITS)

Doppler radar ball on MIT campus, doppler_ball.fits (FITS)

Prudential building, PrudentialCenterMicroObs.fits (FITS)

The Moon, moon_MO_main.fits (FITS)

The Sun, sunmp2two.fits (FITS)

Jupiter, Jupiter072006025114.fits (FITS)

RR Lyrae, RR_LYR.fits (FITS)

Image of the Solar Eclipse

Article on "Giant Dwarf" galaxy

**Physical resources:** None

**Applying mathematical model:** How could we measure the size of an object across the room?

Students are asked to predict the linear size of a range of objects, using the same relationship between angular width, linear width and distance. Students share predictions in a classroom data table Notes: linear size diagrams, linear sizes table

- Objects in common FITS image (1 pixel = 7.9 x 10
^{-4}radians = 0.00079 radians)- Images of students at work: linear size analysis, linear size analysis 2
- Example of written problem-solving process for one object shown in example image, angular_sizes.fits (FITS): linear size solving problems 1, linear size solving problems 2
- Calculations of all other objects in example image, angular_sizes.fits (FITS): linearsize1, linearsize2

- Doppler radar ball on top of MIT Building (1 pixel = 7.9 x 10
^{-4}radians = 0.00079 radians, distance to detector = 174 meters)- Image and student calculation for this problem: doppler ball, doppler ball analysis

- Prudential building in Boston, from MicroObservatory at Harvard (1 pixel = 2.5 x 10
^{-5}radians = 0.000025 radians, distance to detector = 5360 meters)

Applying mathematical model: How could we measure the size of an object across the solar system? Other objects with which to calculate given below:

- Moon, from MicroObservatory main scope (1 pixel = 2.5 x 10
^{-5}radians = 0.000025 radians, distance to detector = 3.8 x 10^{8}meters) - Sun, from MicroObservatory main scope (1 pixel = 2.5 x 10
^{-5}radians = 0.000025 radians, distance to detector = 1.5 x 10^{11}meters) - Jupiter, from MicroObservatory main scope (1 pixel = 2.5 x 10
^{-5}radians = 0.000025 radians, distance to detector = 7.63 x 10^{11}meters) - Stars, image of RR Lyrae from MicroObservatory main scope (1 pixel = 2.5 x 10
^{-5}radians = 0.000025 radians, distance to detector of the nearest possible stars to Earth ~ 5 light years = 4.5 x 10^{16}meters. The stars in this field are most likely much further than this, so this estimate gives a lower limit on the size of stars, by assuming they are very close.)- This calculation leads into the discussion below of minimum detectable angular size, as their predicted linear size will be about 10
^{4}times larger than the sun, and the largest stars are only about 10^{3}solar radii)

- This calculation leads into the discussion below of minimum detectable angular size, as their predicted linear size will be about 10

**Teacher tips/tricks:**

- For one of the above measurements, instructor could have a discussion of measurement error, perhaps calculating an average to see that class does better as a whole, when comparing to the "accepted" value. This is particularly true for objects without well defined edges.

**Assessment ideas:**

- In phrases heard with regard to everyday experience or images, replace words like "size" or "big/small" with correct references to angular diameter, radius, or linear diameter/radius, etc.
- That car looks
**huge**in that picture! - My house is big.
- Compare the size of that building to the one next to it—what a difference!
- This advertising picture of a pizza makes it look a lot bigger than I bet it really is.

- That car looks
- Eclipses: Describe the image of an Earth-based solar eclipse in a sentence, using the words "angular radius", "linear radius" and "distance to the detector". Then, watch a video of a solar eclipse taken by one of the Mars rovers when the Martian moon Deimos passed in front of the sun. How could you change Deimos if you wanted to make the eclipse look like Earth's?
- Read and explain about "giant" dwarf galaxy (A small, nearby galaxy that astronomers thought was much farther away, and thus much bigger.)
- If you doubled distance to an object, what would happen to its angular size in an image? (i.e. using the mathematical model for simple predictions on the level of "two times bigger or half as big".)