1. Exoplanet

An exoplanet or extrasolar planet is a planet that orbits a star other than the Sun. Over 2000 exoplanets

have been discovered since 1988. Specifically, 2098 planets in 1342 planetary systems including 509

multiple planetary systems as of 24 March 2016.

Exoplanets are usually found using very powerful telescopes. However, amateur astronomers can locate

some exoplanets using less powerful telescopes. Using these instruments produces small errors that

can affect the results of the observation, causing the exoplanet to look less bright. When the exoplanet

is viewed through a lens, the error E in magnitude ( brightness) is estimated by the following formula:

E(D, Z, h, T) = 0.09D-273 (27)-1/2(_1

1.75

cos Z

e(-h/8000)

Where D is the diameter, measured in cm, of the lens of the telescope; Z is the zenith angle, in degrees

of the planet, h is the altitude, in meters of the telescope, and T is the exposure time in seconds. The

zenith angle is defined to the angle from the highest point in the sky (Z=0') to an object in the sky.

1.

Suppose the diameter of the lens is 7.1 cm, the zenith angle is 20 and the altitude is 3000m,

what is the minimum exposure time needed to keep the error (E), below 0.001? Round to the

nearest second

2. Repeat Problem 1 for h = 4000m and h = 5000m. Comment on the effect of altitude on the error

3. Find lim E(7.1, Z,3000,180) and lim E(7.1, Z,3000,180) , Comment on the effect of the

z-0

z-,90

zenith angle on the error.

4. Find

OE

JE JE JE

aD ' oz ' oh ' OT

Interpret what these partials represent

5. Find the total differential dE ( what does this represent?)

6. Suppose D=7.1, Z = 20, h =3000, T = 180, use the total differential to determine which of the

following reduces the error by a greater amount:

a) Using a telescope with a lens diameter of 7.3 cm

b) Using a platform to increase the altitude of the telescope to 3200m