Problems on Homework 2

• Calculating S/N

What is the noise for a source with a signal of 1000 countsand a background of 200 counts?

• Are the measured radial velocities in agreement? Why orwhy not? What does it mean to be within 1-sigma?

• Rolling dice problem

What can you say statistically about the “fairness” of thedice? What do you expect? What “error” is expected?

121110987611109876510987654987654387654327654321654321

8%2

0%0

4%1

8%2

28%7

24%6

12%3

8%2

0%0

8%2

0%0

measured

2.8%0.7

5.6%1.4

8.3%2.1

11.1%2.8

13.9%3.5

16.7%4.2

13.9%3.5

11.1%2.8

8.3%2.1

5.6%1.4

2.8%0.7

predicted

12111098765432value

In looking at the dice problem it isimportant to figure out what you wouldexpect.

Mean = 7.36

Median = 7

Mode = 8

48% (12) > 7

28% (7) < 7

Should be 10-11

Light and Radiation

• Light as a wave:– wavelength, frequency, color, energy

c= λν E=hν

Frequency is thenumber of crests(or troughs) thatpass a pointevery second.

The speed of light, c, is constant, 3x108 meters/second

The “Colors” of Light

Color is an expression of the wavelength/frequency/energy of light

How are temperature and color related in adense object?

Blackbody Radiation

Cluster of very bright stars,at 60,000 K, these starsradiate strongly in theultraviolet.

cool gas cloud, temp=60K, emits mostly low-frequency radio radiation.

A dim, young star (red), thestar's atmosphere, at 600K,radiates most in theinfrared.

The sun's surface, at 6000 K,is brightest at visiblewavelengths.

Sagittariusstar cloud

A. Star A gives off morered light and looksredder

B. Star B gives off morered light and looksredder

C. Star A gives off moreblue light and looksredder

D. Star B gives off less redand blue light, but looksredder

MagnitudesMagnitude is a logarithmic measure of brightness, flux is thelinear measure of the amount of energy falling on a given areain a given time. You can subtract fluxes, but not magnitudesdirectly to get a brightness difference.

m = -2.5log(f) + cThe zero point for magnitude is the star Vega

m2 - m1 = -2.5log(f2/f1)A difference of 5 magnitudes is a factor of 100 in brightness

Absolute magnitude (M) is the brightness of an object at 10 pcm-M = 5 log (d/10) = 5log(d) - 5 d in parsecs

Surface brightness

The brightness per unit area for a source - this is only appliedto extended sources, not stars.

If you have 2 equal sized regions with surface brightnesses of21 mag/arcsec2 and 22 mag/arcsec2. What is the averagesurface brightness over both regions?

SB =-2.5log( 0.5*[10-0.4(21) + 10-0.4(22)]) = 21.39

FiltersFilters allow you to look at one section of an objects spectrum at atime. There are several standard filter sets in astronomy.

ColorsThe color of an object is the magnitude difference between twodifferent filters:

B-V = mB - mV

If, for example, the source is a blackbody (most things inastronomy are at least close to blackbodies) then the difference incolor tells you something about the difference in the temperature ofthe blackbody

Telescopes

Light gathering Power:

The amount of light that can be gathered by a telescope in a given amount of time is given by:t1/t2 = (D2/D1)

2

The larger the diameter the smaller the amount of time.

Magnification with eyepiece:

The magnification of a telescope is only useful for extended objects (e.g. the moon, planets,nebulae, galaxies), not unresolved objects such as stars: m = fscope /feyepiece ). Typical eyepiecesare 25 mm, 12.5 mm, and 6 mm. With a telescope of 1 meter focal length these eyepieceswould give magnifications of about x40, x80, and x170 power. This means that objects such asthe moon would appear 40, 80, or 170 larger in diameter (and hence they would appear to be40, 80, or 170 times closer).

F-ratioThe "f-ratio" of a telescope or camera is given by the ratio of the focal length to the apertureand is therefore defined as f/ = f/D. Telescopes with small focal lengths (e.g.,f/2) areconsidered fast while ones with large focal lengths (e.g., f/8) are considered slow.

Image brightness

B ∝ 1/(f/2)The image brightness is inversely proportional to the square of the focal ratio

Limit of Resolution

Diffraction by a circular aperture (such as a telescope objective) results in point source being imagedas a central maximum surrounded by circular rings. The “resolution” of a telescope is given by:sinθ = 1.22 (λ/D)

At this limit the maximum of the diffraction pattern of onestar will fall on the first minimum of the other star.

The atmosphere sets a limit on resolution because the waves do not all follow the same path throughthe atmosphere. This limitation is called “seeing”

Adaptive optics move segments of the telescope mirrorindependently and in real time to account for these traveldifferences through the atmosphere and to, therefore,Improve the image beyond the usual “seeing”

Field of View:

The angular size of the area of the sky that can be imaged on a CCD or viewed through theeyepiece. The field of view is a function of the optics of the eyepiece itself and itsmagnification which is a function of the telescope focal length. Typical eyepieces have field ofview ranging from 40° to 65° or more. The intrinsic eyepiece FOV must then be divided bythe magnification to get the effective field of view at the telescope. A 25 mm eyepiece on a 1meter focal length telescope has a magnification of x40. This will yield a 1° field of view ifthe eyepiece has a 40° intrinsic field of view

Image Scale:CCDs have individual photoreceptor elements with sizes typically as: 9-30 µ = 0.009-0.030 mm atoptical wavelengthsimage scale (arcsec/pixel) = (206.265 µ)/fµ is the pixel scale in microns and f is the focal lengthIn general, we wish to have the pixels mapped to a size whereby they are about two times less than theexpected image resolution delivered by the optics + atmosphere

Nyquist sampling thereom:Optimally sampling all of the information contained in an image requires about two pixels perresolution element. Sampling the resolution finer than this does not yield you more information and canbe considered ``wasteful". �Sampling more coarsely means you are not sensitive to all of the find detailin the picture and you are losing information