Welcome to http://www.handsonuniverse.org/activities/Explorations/Uranus/titania.html

Find Titania.  Measure its distance from Uranus in pixels.  Using the 'image info' and data about Uranus, calculate Titania's orbital radius from Uranus.

Uranus' axis is tilted 97 degrees with respect to its orbit around the Sun. It is tilted so far over that it seems to be rolling on its side. At the time of this image, the North Pole of Uranus was pointing almost right at the Sun (and therefore towards us ....).

Titania revolves in the equatorial plane of Uranus. From our point of view at the time of this image, Titania never went behind or in front of Uranus, but circled around the planet, always staying the same distance away. (Titania's orbital eccentricity is 0.0022.)

A. Which moon is Titania?

Measure Angular Separation on the Image. We can measure angular separation on our image by measuring pixels with slice and multiplying by the pixel scale.

Compare with Published Values. Then, we can look up the angular separation of Uranus and Titania on the date this image was taken, and compare the published separation with our measurement on the image.

Measure Angular Separation on the Image.

On the Uranus.fts image, measure distance in pixels (picture elements). Using slice, find the number of pixels from the center of the moon you think is Titania to the center of Uranus.

What is the distance you measured in pixels? _____________

Compute Angular Separation

Next, convert pixel distance to angular separation. Pixel Scale on an image tells the angular size of the sky represented by each pixel. (Image info under Data Tools: Xscale and Yscale tell the pixel scale values.) The pixel scale for this Uranus.fts image is 0.6 arcseconds/pixel. Multiply the distance measured in pixels by 0.6 arcseconds/pixel.

Angular Separation: What is your answer in arcseconds? __________

The Astronomical Almanac, published by the United States Naval Observatory, lists the orbital parameters of all the Planets and their known moons each year. According to the Almanac on Dec. 20, 1997 (UT), the angular separation between Uranus and Titania was 20 arcseconds.

Compare with Published Separation

If your answer is close to 20 arcseconds, the object you selected is probably Titania! If you are less than 20 arcseconds, you may be measuring from Ariel or Umbriel. If your answer is more than 20 arcseconds you are probably measuring from a star in the image!

B. What is the distance between Titania and Uranus in kilometers, miles or AU?

How do we convert angular size in arcseconds to real size in kilometers, miles or astronomical units (AU)? (An astronomical unit is the distance from Earth to the Sun.)

We will use the published distance from Earth to Uranus, trigonometry and a calculator. (Trigonometry is mathematics using the numerical properties of right triangles).

1. Gather Known Information.

   According to the Astronomical Almanac, on Dec. 20, 1997 (UT), the distance between the Earth and Uranus was 20.6 AUs. (One AU = 93 million miles or 150 million kilometers.)

   In part A you measured the pixel separation of Titania and Uranus and then calculated the angular separation between them.

   What was the value you calculated above for the angular separation between Uranus and Titania? ________________

   
   This is a picture of what we know. The distance D is what we want to find out.

1. Using Trigonometry and a Calculator solve for the Unknown Distance

Titania, Uranus, and Earth form a right triangle. Using principles of trigonometry, the tangent of the angle Uranus-Earth-Titania (which is also their angular separation) is equal to the distance between Titania and Uranus (D) divided by the distance between Earth and Uranus (20.6 AU). With a calculator, this is easily figured out.

1. Change angular separation to degrees by dividing by 3600 (60 arcseconds per arcminute times 60 arcminutes per degree).

2. Take the tangent of the result. This will be a very, very small number!

3. Multiply by 20.6 AU, then by either 93 million miles or 150 million km per AU.

The result is the distance from Uranus to its moon, Titania.

What is your result for the distance between Uranus and Titania? __________________ (Hopefully your answer is around 300,000 kilometers or 180,000 miles. This is the published distance between Uranus and Titania.)

C. Challenge: Work out the distances between Uranus and Umbriel or Ariel.  Apply the same procedure for Uranus' other two moons!

Back to Uranus and Its Moons

01/09/2000