A faint spectroscopic quasar survey

C o-S upervisors D r. F . D. A. H artw ick

D r. D. C ra m p to n

A very fa in t spectrosco pic q u a sa r survey w as u n d e rta k e n in o rd e r to define th e s ta tis tic a l b eh av io r of th e low lu m in o sity p o rtio n of th e q u a s a r p o p u la tio n a t high re d sh ift a n d to search for evidence o f a decline in space density. B o th slitless a n d m u lti-a p e rtu re spectroscop ic o b serv atio ns w ere o b ta in e d for a p p ro x im a te ly 750 o b ­ je cts w ith B m a g n itu d e s in th e ran g e 20 < B < 24. T h e to ta l n u m b e r o f em ission line c a n d id a te s found w as ~ 50 a n d th e re are 5 excellent q u a s a r c a n d id a te s, 3 very p ro b a b ly a t h ig h red sh ift. In a d d itio n to th e q u a sa r c a n d id a te s a n u m b e r o f em is­ sion line galaxies have b een identified.

E x am iners:

C o -S u p erv iso r Dr. F . D. A. H artw ick C o -S u p erv iso r D r. D av id C ra m p to n

E x te rn a l E x am in er D r. P. S. O sm er

D rT ^/X T B urk^^^ D r. R. R . D avidso n

I l l

T able o f C o n te n ts

A b s t r a c t ii T a b l e o f C o n t e n t s iii L is t o f F i g u r e s v L is t o f T a b le s v ii A c k n o w le d g e m e n ts v iii D e d i c a t i o n ix C h a p t e r 1 I n t r o d u c t i o n I C h a p t e r 2 T h e 1; V a m e t h o d 16 C h a p t e r 3 O b s e r v a t i o n s 34 C h a p t e r 4 D a t a r e d u c t i o n 44 C h a p t e r 5 D a t a a n a l y s i s 73

C h a p t e r 6 R e s u l t s 10 0

C h a p t e r 7 D is c u s s io n 124

V

L ist o f F ig u res

F igure 1.1 ... F igure 1.2 ... F igure 1.3 ... F igure 4.1 ... Figure 4.2 ... F igure 4.3 ... Figure 4.4 ... F igure 4.5 ... F igu re 4.6 ... F igure 4.7 ... F igure 4.8 ... F ig ure 4.9 ... F igure 4.10 ... F igu re 4.11 ... F ig ure 4.12 ... F igure 5.1 ... F ig ure 5.2 ... F ig ure 5.3 ... F ig ure 5.4 ... F ig ure 5.5 ... F ig ure 5.6 ... F ig ure 5.7 ... 12 13 14 54 55 57 59 63 66 67 68 69 70 71 72 79 80 81 86 87 89 93

F igure 5.8 ... 95 F igure 5.9 ... 98 Figure 5.10 ... 99 F igure 6.1 ... 105 F igure 0.2 ... 106 F igu re 6.3 ... 107 Figure 6.4 ... 108 F igure 6.5 ... 109 F igure 6.6 ... 118 F igure 6.7 ... 119 F igure 6.8 ... 121 F ig ure 6.9 ... 122 F igure 7.1 ... 135 F igure 7.2 ... 137 F igure 7.3 ... 139 F igure 7.4 ... 140 F igure 7.5 ... 141 F igure 7.6 ... 144

L ist o f T ables

V I I Table 3.1 ... 35 Table 3.2 ... 37 T a b l e 6 . 1 ... t IT T able 6.2 ... 115 Table 6.3 ... 116 T able 6.4 ... 117

A cknow ledgem ents

It is a pleasure to thank Dr. P. D. A. H artw ick and D r. D. C ram p to n for th eir en thu siasm and encouragem ent. T h an k s also to th e other m em bers of my co m m ittee and to Dr. E. Ellingson for her help a t the telescope. This work is based on observations o b tain ed as a V isiting A stronom er at th e C anada-F rance-Iiaw aii telescope a n d I acknowledge th e su p p o rt of th e N atu ral Sciences an d E ngineering R esearch Council of C an ad a in the form of a P o st-g rad u ate scholarship. T h e D om inion A strophysical O bservatory an d th e C an ad ian A stronom y D a ta C enter provided co m p u ter facilities for w hich I am very grateful.

In tr o d u c tio n

O ne of th e reasons th a t d ista n t quasars are one of th e m ost in te re st­ ing populations of ob jects in th e U niverse is th a t th ey are so lum inous th a t th ey can be observed as th ey were when th e U niverse was ~ 10% of its p resen t age. Surprisingly, we do not see large differences betw een th e detailed sp e c tra of individual ob jects observed locally and those which existed early in th e h isto ry of th e U niverse. T h e m ost strik in g change th a t is observed in quasars is th e change in th e sta tistic a l p ro p erties of th e p op ulatio n. Q uasars were once m uch m ore num erous th a n th ey presently are and this fact is th e m ost obvious evidence th a t we have th a t a po pu latio n of o b jects in th e U niverse has evolved su b stan tially since th e tim e of th e decoupling of m a tte r and rad iatio n .

It is generally believed th a t quasars are th e resu lt of activ ity in th e nuclei of galaxies involving th e release of energy from gas which is falling onto a very m assive o b ject, pro bab ly a black hole (see th e review by Rees 1984), and th e em ission of energy in th e form of light accom panied by tu rb u le n t m otions of gas clouds in th e vicinity of th e galaxy cen ter. T h is belief is m o tiv ated by observations of nearby active galactic nuclei which are clearly associated w ith galaxies (O sterb rock and M athew s 1986) an d which ap p ear to be fueled by a sim ilar (or identical) process to th a t fueling m ore d is ta n t quasars whose presum ed un derlying galaxy is too faint to be observed. N ot only are quasars an in terestin g physical phenom enon in them selves b u t th e ir connection to

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galaxies leads us to believe th a t an u n d erstan d in g of the processes which drive th e evolution of th e space density of quasars as th e U niverse ages will c o n trib u te su bstantially to an u n d erstan d in g of the evolutionary processes which govern galaxies th em selv es particu larly during the early stages of th e ir existence w hich are especially m ysterious.

Since th e ir discovery in th e early 1960’s (M atth ew s and Sand age 1963) an enorm ous effort has been d irected a t quasars in order to try to exploit th em as tools for cosm ological stu dies, although it m ight be said th a t we have learned less ab o u t cosmology, and m ore ab out quasars, th a n was initially ex p ected . On th e o th e r hand, su b stan tial use has been m ad e of quasars as beacons th a t light up intergalactic or p reg alactic clouds so th a t they can be d etecte d and studied (Sargent, et al. 1980, Steidel and S argent .1987). T his is one of th e m ost in teresting uses of quasars .

T his p aper continues th e stu d y of th e space den sity of quasars and th e ir evolution th a t began sh ortly after th e ir discovery (S chm idt 1968) and has progressed to th e point w here i detailed u n d erstan d in g of th e evolution of the qu asar space d en sity now exists from th e present up to

z = 2,2 (Boyle et al. 1987, 198S). T h e behavior of th e space density at

higher redsh ift is less well-defined (S ch m id t 1987, W arren and Hewett, 1990) an d recent surveys (W arren, H ew ett, and O sm er 1988, S chm idt, Schneider, and G unn 1988) continue to m ake rapid progress toward a d eterm in a tio n of th e evolution a t higher redshift and correspondingly earlier tim es in th e history of th e Universe.

i ■i

1.1 Q u asar S u rv ey s

T he space density of QSOs ( “qu asi-stellar o b je c ts”— a term which will be used interch ang eably w ith “quasars” ) is th e n u m b er of ob jects p er unit volum e, w hereas th e lum inosity fu nctio n (L F ) is th e n u m ­ ber of o b jects p er u n it volum e p e r m ag n itu d e interval. T he L F can be observed at various redshifts; tak in g a slice a t fixed redshift across lum inosity gives us th e LF a t th a t redshift. Slices can be taken a t dif­ ferent redshifts a n d these slices can be com pared w ith one an o th er to define th e evolution of th e lum inosity function. H ere evolution refers to th e change w ith red sh ift of the lum inosity fu n ctio n . M any m eth o d s to co m p u te th e L F are available, som e assum e th e valid ity of a p a rtic ­ ular m a th e m a tic a l m odel a n d fit th a t m o d el’s p a ra m e te rs against the observations, an ap p roach th a t was th e no rm in early work w hen th e observations were very incom plete (S chm idt 1968). It is now possible to d eterm in e th e L F w ith no assum ptions w hatsoever a n d get m e a n ­ ingful answ ers—obviously a n approach superior to m odel fittin g unless one has som e physical basis for believing th a t a p a rtic u la r m odel is correct. T h e reason th a t th is has becom e a p ractical possibility is th a t g re a t advances have been m ad e in surveying th e sky using a n u m b er of different observing techniques.

Regardless of th e m e th o d th a t is used to co m p u te th e LF from th e d a ta , it is essential to begin w ith d a ta whose selection c rite ria are well- defined. In o th er w ords, th e parent p o p u latio n can b e reco n stru cted (or estim a te d ) from th e d a ta only if one u n d erstan d s th e selection effects or sam pling function of o n e’s survey technique. T his has tu rn ed o u t to

4

be a trick y problem in th e case of quasars an d especially so in th e case of surveys which d etect quasars on th e basis of em ission lines rath er th a n colors or radio em ission because the em ission lines them selves display a large range in stren g th . T h e colors of quasars also show a broad d istrib u tio n as does th e ratio of raclio-to-optical lum inosity; still th e em ission line selection effects are the m o st difficult and it is only recently th a t com prehensive tre a tm e n ts of th e em ission line- jn-oblem have ap p eared (S chm idt, Schneider, and G u n n 1986a., G ra tto n and O sm er 1987, and §2 of this paper).

Q uasars are selected by th e ir peculiar energy d istrib u tio n s which dif­ fer m arkedly from those of norm al stars. T he selection m ay be m ade a t very low sp ectral resolution using broad-band filters (e.g. £/, B , V, R, / ) , w ith typ ical w idths of 800

A,

or a t m o d e ra te resolution using spec­ troscopy w ith resolutions of ~ 50 to

100A.

B ro ad -b an d filters sam ple th e co ntinu um energy d istrib u tio n w hereas spectroscopy selects by th e presence of em ission lines. T h ere is an advantage in principle to the use of em ission lines to segregate quasars from th e stellar and galaxy pop­ ulations; nam ely th a t, once broad em ission lines are found, it is certain th a t th e o b ject is a quasar. This is a m a tte r of definition and fur­ th e r observations m ay be needed only to confirm the o b je c t’s redshift, w hereas an object selected because of its broad-band colors m u st still be observed spectroscopically in order to determ in e w h eth er or n o t it is a q u asar. P a rtic u la rly a t faint m agn itud es, any reduction in th e q u an tity of follow-up observations required is a huge advantage because such ob­ servations consum e considerable am o u n ts of large telescope tim e. T he

advantages of em ission line surveys can be m ade alm o st overw helm ing if th e survey is designed so th a t it provides not sim ply a c a n d id a te list but unam biguous identifications and sim ultaneously gives p h o to m e try and redshifts for th e quasars. T h en th e survey is a one-step process and can be com pleted in a single observing season and th e results published quickly. T his goal has been approached by th e p resen t survey b u t no t q u ite achieved. M odifications to th e procedure used here should m ake this am bitious goal attain a b le.

To begin, a brief discussion of cu rren t optical surveys is given and th en these results are discussed in term s of th e c u rren t s ta te of know l­ edge of th e lu m inosity function .

1.2 S u r v e y s b a sed on color

Q uasars were originally discovered as a d istin ct class of o ptical ob­ je c ts d u rin g th e process of identifying radio sources (M atth ew s and S andage 1963, S andage 1965). It rap idly becam e evident th a t th e ra ­ dio sources were often ex trem ely blue o b jects and th a t m a n y objects w ith sim ilar b u t unfam iliar colors existed which were n ot rad io sources. It is now known th a t radio-quiet quasars o u tn u m b er radio-loud objects by a large factor ( ~ 10 to 100, W eedm an 1986).

O nce it was realized th a t this new and in terestin g class of astro n o m ­ ical o b jects could be readily d etected by th e ir un usual colors, optical searches were begun (Sandage and Veron 1965), an d ex isting catalogs of very blue o b jects (e.g., H aro a n d L uyten 1962) were searched for likely can d id ates. Braccesi et al. (1980) give a clear descrip tion and

6

illu stratio n s of th e ir m ethod of q uasar selection. Very nearly com plete sam ples of quasars a t z < 2.2 can be ob tain ed by selecting objects by their U — B colors and, although th e re is su b stan tial co ntam ination at b rig h ter ap p a re n t m ag nitudes, only ~ 15% of th e ob jects selected are not Q SO s at B > IS (Braccesi et al. 1980). U ltraviolet excess selection (U VX) is an efficient technique because it requires observations in only two bandpasses and d em o n strates an excellent level of com pleteness (very few tru e qu asars are m issed) as well as a high success ra te (very few of th e cand id ates selected by this m eth o d are co n tam in a n ts).

Because it is such a reliable m eth o d , the selection by UVX (large negative U — B color) has been m uch applied (Braccesi et al. 1980, Schm idt and G reen 1983) along w ith m odifications such as th e inclusion of a B — V color. O ne of th e largest surveys ever done is th e U V X survey of Boyle e t al. (1987,19S8) th a t provides a m ajor co ntrib ution to our p resen t u n d erstan d in g of the lum inosity function,

A lim ita tio n of th e U V X m e th o d is th a t it is not effective a t d e te c t­ ing quasars at red shifts g reater th a n a b o u t 2.2. Q u asar colors change w ith red shift (G rew ing 1967, Veron 19S3) as em ission lines an d co ntin ­ uu m featu res are shifted into and o ut of th e observing bandpasses. T his m ay resu lt in m o d e ra te levels of incom pleteness at z < 2.2 (G reen 1989) b u t resu lts in very low d etectio n rates a t z > 2.2 w here the stro n g Lya. em ission line of hydrogen moves from th e U to the B band resulting in an effective loss of th e UV excess.

In rece n t years, color selection techniques have evolved to th e point w here high red sh ift QSOs can be d etecte d with a high degree of success

(Koo and Kron 1988; W arren, H ew ett,and O sm er 1988; W arren an d H ew ett 1990). In order to do this, observations are needed in several bandpasses (4 in th e case of Koo and Kron 1988, and 5 in th e case of W arren, H ew ett, and O sm er 1988) so th a t th e observing tim e for faint objects is considerable. M ulticolor techniques rely on th e segregation in m ulticolor space of quasars from a well defined stellar locus.

M ulti-color p h o to m etric techniques have played a m a jo r rols in as­ sem bling large sam ples of quasars and have been modified to m eet th e challenge of detectin g high redshift ob jects. D irect p h o to m etry will al­ ways be able to reach fainter lim iting m ag n itu d es th a n spectroscopic surveys b u t the fear exists th a t some sm all fraction of q uasars will be lost in th e stellar locus.

1.3 E m issio n lin e su r v e y s

Hoag and Schroeder (1970) m ade th e first observations of qu asar em ission lines w ith a survey ty p e (slitless) in stru m e n t and showed th a t th e rem arkably conspicuous spectroscopic ap p earance of quasars could be exploited for survey purposes. A nu m b er of ensuing surveys quickly d em o n strated th e effectiveness of this tech niqu e (S m ith 1975, Hoag 1976) an d in terest was stim u lated by th e fact th a t th e stro ng est line in a typical quasar sp e c tru m ( L y a ) ap pears in th e o p tical band a t z ~ 2 an d rem ains in th e o p tical until z ~ 5. A ided by th e cosm ological (1 -1- z) boost factor of th e equivalent w idth, th e p o te n tia l for probing th e very d ista n t U niverse is extrem ely a ttra c tiv e .

(charge-8

coupled device) an d O sm er (1982) used a grism -plate com bination in an a tte m p t to find very high redshift (z > 4) ob jects. Both of these a tte m p ts failed to find very high redshift quasars and th e O sm er (1982) resu lt provided a stro n g suggestion th a t th e space density a t high red­ shift was lower th a n a t z ~ 2. T h e question of com pleteness of emission line surveys was raised by Clowes (1981) and has been dealt with by S ch m id t, Schneider and G unn (1986a) and G ra tto n and O sm er (1987). T h ese stu d ies have shown th a t, although em ission line sam ples may b e biased tow ard o b jects w ith large em ission lines, the effect can be acc u rately quantified.

Spectroscopic surveys p resently underw ay or recently com pleted in­ clude th e Large B righ t Q uasar Survey (Foltz e t al. 1987, 19S9) the large C F H T survey (C ram p to n , Cowley, and H artw ick 1989,1990) and th e CCD surveys of S chm idt, Schneider, an d G unn (1986a,1986b,19S8). T hese surveys have provided sam ples th a t are co m plem entary to the color-basecl survey sam ples, and have ex ten d ed th e redshift range over w hich very large sam ples are available. T h e a ttra c tiv e feature of spec­ troscopic surveys is th a t they select ob jects using c rite ria by which quasars are actually defined— th a t is, th e spectroscopic app earan ce of th e objects.

Q uasars can also be d etecte d by th e ir variability (U sher e t al. 1978, Trevese et al, 1989), by th eir lack of proper m otion (S andage and L u yten 1967, K ron and Chiu 1981), or by radio, X-ray or infrared em ission; b u t none of these techniques tu rn s up a quasar population w hich escapes o p tical detection . T he optical sam ples are th e m ost

com plete and th u s they are th e sam ples which are m o st su ita b le for d eterm in a tio n s of the space density of the q uasar popu lation .

1.4 T h e lu m in o sity fu n c tio n

T he tw o essential ingredients th a t are needed to specify th e behav­ ior of th e lum inosity function are th e survey d a ta a n d a m eth o d of analyzing these d a ta . A p ro p er analysis requires an u n d e rsta n d in g of th e selection effects in th e d a ta an d a m eans of correcting for th e m in order to o b ta in an unbiased estim a te of the q u asar lum inosity function

T h e survey d a ta is now su b stan tial and covers m uch of th e [M,z] plane although n o t uniformly. T h e m ethods available to analyze th e d a ta have evolved in step w ith th e survey d a ta them selves. O ne of th e rem ark ab le achievem ents in th e analysis of q u asar d a ta was by S chm idt (1968), w ho co rrectly ou tlined the density increase an d decline in th e qu asar d en sity based on a very lim ited sam ple of objects. T his was done using th e so-called V f V max m eth o d (S chm idt 1968, S chm id t and G reen 1983) w hich fits th e d a ta to find p aram eters of an assum ed density variation m odel.

A n u m b e r of procedures have been applied to th e p roblem of the q u asar lu m inosity function an d the variation of space d en sity w ith red­ sh ift (S chm idt 1970), including n o n -p aram etric sta tistic a l techniques (T u rn er 1979) and m axim u m likelihood m odel fittin g (M arshall et al. 19S3). T h e discussion has revolved around tw o fu n d am e n tal m odels: p u re den sity evolution (P D E ), for which th e shape of th e lum inosity

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function rem ains unchanged with redshift while its am p litu d e varies; and p u re lum inosity evolution (P L E ), for which th e sh ape of the LE rem ain s unchanged b u t th e d istrib u tio n moves uniform ly tow ard higher lum inosity w ith increasing redshift, as if each individual o b je ct in the lu m in o sity function were becom ing b righ ter w ith redshift a t an identical rate . D epending upon one’s inclination, these models may be thou gh t of as co ntain in g physical inform ation ab o u t the behavior of individual quasars, or sim ply a,s convenient descriptions of th e evolution of th e sta tis tic a l p ro p erties of the q uasar population.

1 .4 .1 T h e lo w r e d s h ift lu m in o s ity fu n c tio n

For a nu m b er of years it had been shown th a t the observations of z < 2 quasars were consistent w ith one or both of the fu nd am en­ ta l m odels (P L E an d P D E ) of evolution of th e lum inosity function (B raccesi e t al. 1980, M arshall et al. 1983, Koo and Kron 1982) but progress was m ade w ith m odifications to th e basic m odels by S chm idt an d G reen (1983) and C ram p to n , Cowley, and H artw ick (1987). T h e present “sta n d a rd m odel” of the evolution of the lum inosity function a t z < 2.2 resulted from a m ajo r step forward in th e observational field, Boyle et al. (1987,1988) presented results from a large UV-exccss survey ( ~ 400 o bjects) and showed th a t these new observations are co nsistent w ith th e pure lum inosity evolution model of th e q u asar pop­ u lation for 2 < 2.2, Figure 1.1 (ad ap ted from H artw ick and Schade 1990) shows th e resu lts for a com bined sam ple of quasars from all of th e m a jo r surveys. It is easy to im agine th e lum inosity function sliding along th e m a g n itu d e axis tow ard higher lum inosity from low to high

redsh ift.

1 .4 .2 T h e h ig h r e d s h ift lu m in o s ity f u n c tio n

As m ore d a ta have becom e available from surveys th a t sam ple the high redshift qu asar population, it has becom e possible to .ast w h ether an ex trap o latio n of th e pu re lum inosity evolution m odel tow ard higher redshift agrees w ith th e oservations. T h e C F H T survey (C ram p to n , Cowley, and H artw ick 1989) provides th e largest sam ple to fain t m ag­ nitudes w hereas sm aller sam ples from S chm idt, Schneider, and G unn (1986a,1986b,1988) and W arren, H ew ett, and O sm er (1988) p ro b e th e very high redshift region.

T h e d a ta are n ot yet as com plete as one w ould like b u t th e re seems to exist a decline in space density a t 2 > 2, (p o in ted o u t by O sm er 1982;, also see G reen 1989) and this decline is incon sisten t w ith th e P L E ev olution ary m odel. F igu re 1.2 (ad ap ted from H artw ick an d Schade 1990) shows the lum inosity function a t high redshift for a com bined sam p le of quasars. T h e d etails of th e ap p are n t decline are n ot clear.

T his ap p a re n t decline in th e space density of quasars has been de­ fined largely by observations of m o d erately b rig h t quasars (W arren, H ew ett, and O sm er 19S8, S chm idt 1988) w hereas th e behavior of the po pu latio n of faint quasars (which com prise th e overw helm ing m a jo r­ ity of th e p o p u latio n ) a t high redshift is unknow n. F igure 1.3 shows th e behavior of th e space density of m o d erately brig h t QSOs w ith red­ shift. (T h e space density is th e in teg ral of th e lum inosity functio n.) It ap p ears th a t th e re is a decline in density following a poorly defined m axim um n ear 2 = 2.

a = —0.5

a

a. o

•e*

0.4 < z < 0.7 L ocal QSOs (z<0.2) S e y fe rt 1 g a la x ie s

- 2

- 2 6

- 3 0

- 2 5

- 2 7

- 2 8

- 2 9

- 2 4

- 2 3

- 2 2

M

b

Figure 1.1 T he lum inosity function is shown for a number o f redshift slices (adapted from results Hartwick and Schade 1990). It is easy to imagine the lum inosity function sliding along th e M axis toward higher lum inosities as the redshift increases. T he Seyfert 1 and local QSO lum inosity function form a coherent part o f this picture of th e evolution o f the quasar population.

— 3

L

og

$

G

pc

m

a

g

0 2.2 < z < 2.5 2.5 < z < 3.3 _\. \ 4 (WH m o d e l) z I I I I I I I I I I I I I I I I I I I I

-I J—1 I I I I I I I I I

—22

- 2 3

- 2 4

- 2 5

- 2 6

- 2 7

- 2 8

M

b

- 2 9

i i

- 3 0

Figure 1.2 T he high redshift lum inosity function is illustrated (from th e com pilation o f IIS) along with the esti­ m ated lum inosity function at z — 4 from Warren and H ewett (1990). T h e change in the lum inosity function with redshift is not as dram atic as it is at lower redshift (Figure 1.1).

1000

800

CO 1 O cu o

600

CD C\2 II

S 4 00

v

20

0

0 1

2

3

4

R edshift z

Figure 1.3 The integrated space density (QSOs brighter than M = - 2 6 ) changes radically but the high red­ shift uncertainties are very substantial. The dotted lines connecting open circles represent the parameter set

’q0. o ’= [0 .1 0 .-1-0] and the solid line with filled circles assumed [0.50,-0.50]. The qualitative behavior does not

depend on these parameters. The filled triangles represent th e results o f Warren and Hewett (1990) and assum e [<7Q. a ] = ' 0 . 5 . - 0 . 5 ] .

The present work is an a tte m p t to define the behavior of th e low lum inosity sources a t m o d erate to very high redshift. T h e ex isting d a ta are fragm entary for th e faint end of th e lum inosity function a t z > 2, which m akes the ta sk m ore difficult. O ne would like to sam p le th e en tire red shift range of in terest (2 < z < 4) w ith a single survey to im prove th e d eterm in a tio n of th e faint end of th e lum inosity function a t z ~ 2 as well as define th e lum inosity function a t very high redshift

C h a p te r 2

T h e 1 / V a m e th o d

1 6

T h e m ost basic way to co m p ute the space density of any population is sim ply to ta k e each o b ject in tu rn an d d eterm in e w hat volum e of space had been searched in o rder to find it. T h a t o b je c t’s co n trib u tio n to th e space density is then th e inverse of the volum e searched (th e “accessible volum e” of Avni an d Bahcall 1980). T h e following work does ex actly th is and is related to results of Choloniewski (1987) th a t m odified th e “C -m eth o d ” of Lynden-B ell (1971). P a rts of th is section are from Schade (1988).

T h e following sections show how to find the w eighting th a t is ap p ro ­ p ria te for each o b je ct in a sam ple w ith well-defined selection lim its in o rd er to convert observed counts to a space density or lum inosity func­ tion. T h e only tricks which are required for co n stru ctin g these relations are two p ro p erties of th e d e lta function:

CO

(0

and 00 (2) — OO

a n d a p ro p erty of th e H eaviside function

CO CO

These two special functions form the h e a rt of th e description of th e lum inosity function; th e d e lta function because it describes th e actu al observations of discrete ob jects, a n d th e H eaviside fun ctio n because it represents the selection function (its arg u m en t m u st accu rately define w hether an o b je ct is d etected or n o t d etecte d ). N ote th a t th is im plies th a t the p ro b ab ility of d etectin g an o bject can ta k e on one of two val­ ues: 0 or 1. In reality, given an o b je ct w ith a specific tru e em ission line flux, the observed flux a n d thus th e d etectio n p ro b ab ility is governed by photon sta tistic s (i.e. Poisson statistic s). T hus th e d etectio n p rob a­ bility (as a function of em ission line flux) does not display a sh arp edge. T h is effect can also be included (G ra tto n a n d O sm er 1987).

2.1 M a g n itu d e lim ite d sa m p le s

A m ag n itu d e lim ited sam ple is one whose m em bers have been chosen on th e basis of ap p a re n t m a g n itu d e and such a sam ple is assum ed to be 100% com plete for all objects b rig h ter th a n its lim itin g m agn itud e.

Assum e th a t th e lum inosity fun ction $ ( M , z) can b e represented by a sum o f w eighted d e lta functions as can th e observed d istrib u tio n . Consider a lim ited red shift range Z\ to z2 over w hich th e lum inosity function is ap p ro x im ately co n stan t w ith z a n d so is a function of abso­ lu te m agn itu de only.

N i2(M ,z)d zd M = $ (M )d V d M 0 (m u m - m ( M ,z ) ) 0 ( z 2 — z ) 0 ( z — z i) (4)

N 12( M , z ) is th e observed d istrib u tio n and 0 (a\) is th e H eaviside function which equals 1 if x > 0 an d equals 0 otherw ise. I will define:

18

In term s of d elta functions th e above equation is: clV dz £ S ( M - Mi, z - Zi) = J 2 tp]S(M - M j)0 z“ -0 (m u m - m (M , z)) (5) where : S ( M ) = £ * S(M - Mj) (C) J Z j 6 ( z a , 2 2 )

by assu m p tio n and th e su m m ation indices refer to individual o b ­ jects.

Define z™ax as th e m axim um reclshift at which th e n l/l o b ject would still be d e te c ta b le w ith th e given lim iting m a g n itu d e mii,n:

m (M n, z“ ax) = miim. (7)

In teg ra tio n of equ ation (5) over all redshift z gives th e following result:

_ _ _ _ °r d V

^ ( M - M i ) = X ; ^ ( M - M j ) J 0 ( m liin- m ( M , z ) ) 0 a— dz. (8)

1 0

zi6(z1,z2) zjS(zi,z2)

T h e H eaviside functions can be accounted for in th e lim its of th e integral. T h e resu lt is:

m in (z 2lzjnaK) r ci V £ 6 ( M - M 0 = £ ^ ( M - M j ) J — dz (9) *te(»i.*2) zje(zi,z2) Zl 2 « ( M - M 0 = 2 > j « ( M - M , ) V i (10) i i Z|€(zi,Z2) aj€(aj,*2) w ith:

(1 1)

so th at:

( 12)

VJa is th e volum e accessible to th e }th o b je ct in th e red sh ift interv al un der consideration. Im plicit above is th a t th e volum e elem ent dV takes account of th e sky coverage of th e survey.

T h e m e th o d m ay be generalized to several sam ples. Now th e d if­ ferent sky coverages will be accounted for by defining u) = Cl/Air w ith H the sky coverage in sterad ian s an d dV th e volum e elem ent over th e w hole sky.

A ssum e th a t we are still looking at a lim ited redshift range a n d th a t eq u atio n (6) holds w ith th e sum ru n n in g over all o bjects in ( z i ,z 2) regardless of th eir sam ple of origin. In ad d itio n to different sky cover­ age, the different sam ples have different m a g n itu d e and redsh ift lim its w ithin w hich quasars quasars can b e d etecte d . T h e a ih sam ple is sen ­ sitiv e to o b jects which m eet th e following conditions:

< z < z" m ? < m < m “

an d define 0 a = l if all of these conditions are m e t and 0 “ =O otherw ise. T h e observed d istrib u tio n of ob jects on ( z i ,z 2) is th e su m of th e c o n trib u tio n s from ea.ch sam ple:

N 12(M , z)dzdM = X ) $ ( M K d V d M 0 a 0 z (13)

20

which in te rm s of d e lta functions is:

M - M SW Pia P\ dz

£ « ( M - M| ,z - zi) = - M i K 0 " e , ^ . (W)

i a j

» i 6 ( * l . s s 2 ) »j e ( z i , 2 2 )

In teg ra tio n over redshift gives:

00 IV

2 2 « ( M - M , ) = x ; * « ( M - M j ) x ; w„ / e ° e , ^ - d z . ( 15)

i j a i C l Z

« i € ( * i »Z2) zj € ( z i .*2 )

T h e H eaviside functions can be expressed as lim its on th e integral. in in (z 2,z j,z ||,jax)

I \ /

= / c- [iz. f ie )

1 j ex Cl 55

*16(*i ,a2)

*(€(*1.32)

max(zi,z“,z",lln)

w ith z™jax defined analogously to equ ation (4). Define:

min(z2,z“,z£'jax)

( l 7 )

max(zi,zJ,z“jln)

Then: 2 > ( M - M , ) = E {(M - Mi ) ' A i E v r (IS) i j of zi€(zi,z2) Zj6(»i,32) and

*-fW-

_a

(l9)

T his resu lt sta te s th a t ?/>j is th e reciprocal of th e sum of th e volum es accessible to th e j th o b je ct in all sam ples, not only in th e sam ple in which it was discovered. T hus th e com bined sam ples are tre a te d as a “c o h ere n t” sam p le in th e term inology of Avni and Bahcall (1983). T h e

sam ples m ust be com bined in such a m anner th a t no specific volum e of space is surveyed m ore th a n once.

T he lum inosity function is converted from a sum of d e lta functions by averaging over m a g n itu d e bins:

m2 f $ (M )d M

W )

=

(2°)

/ dM Mi . M2 <S(M) ” AM / “ MJ)dM <21> Mi J ^ = £ m ? * s- (22) Mj G ( M j , M 2 )

2.2 E m issio n lin e su r v e y s

T he above resu lts are valid for th e case w here a survey has a single lim iting m a g n itu d e in d ep en d en t of redshift. T h e situ a tio n for surveys w hich select can d id ates because of th e presence of em ission lines is m ore com p licated b ut can be approached in a m a n n er sim ilar to above. It is necessary to have som e knowledge of th e equivalent w id th d istrib u tio n an d it will be necessary to assum e th a t th e equivalent w id th (E ) dis­ trib u tio n is in depen dent of redshift although th is is n ot a fu n d am en tal req u irem en t of this m e th o d cf analysis: th e re is ju s t n o t enough d a ta available to define th e equivalent w idth d istrib u tio n as a function of redshift. In any case this assu m p tio n is reasonable as far as is p resently known. I will also assum e th a t th e equivalent w idth d istrib u tio n is in­ dep en d en t of lum inosity, an assum ptio n which is known to be false b u t

which has little effect in th e p resent case. (T he lum inosity dependence, known as th e Baldw in effect, is weak). F u rth erm o re, one can easily cor­ rect for th e lum inosity effect and th ere is enough inform ation available to define th e lum inosity dependence. It is not necessary in the present case to use this info rm atio n because th e can did ates of interest span a sm all rang e in lum inosity.

It will be helpful at this p o in t to briefly describe th e d a ta th a t will be analyzed w ith th e m eth o d s developed in th e following sections. T h ere are two sets of observations. T h e first is a set of CCD frames (each fram e covering a sm all p atch of sky) ob tain ed in O ctob er 1989 using a grism in slitless m ode. T his m eans th a t no a p ertu res were used and a sp e c tru m for every o b je ct in th e field of view was recorded. T h e second set of observations consists of a n u m b er of CCD frames using th e sam e telescope, grism , and CCD b u t using ap ertu res rath e r th a n slitless m ode. W hen a p ertu res are in tro d u ced this com plicates the d a ta analysis in several ways. In th e first place, sp ectra are recorded only for th o se o b je cts for w hom an a p e rtu re has d elib erately been cut: only a fractio n of th e ob jects w ith in each field are observed. A nother difference is th a t som e light from th e can d id ate o b jects is lost because of th e finite a p e rtu re size w hereas in slitless m ode all of th e light falls on the chip. T h e d a ta analysis m u st tak e these effects into account.

This section will be developed in th e context of a slitless sp ectro ­ scopic survey (th e sim pler case) and then th e necessary changes re­ quired for th e case of a p e rtu re spectroscopy will be m ade.

of CCD frames (each fram e identified by its index k) w ith possibly vary­ ing sensitivities. Each fram e m ay have its own sensitivity zero point

(Lf.) and could have its own sensitivity curve. I t will b e assum ed th a t

th e equivalent w id th d istrib u tio n is \'( E ) and is known. T h e observed d istrib u tio n of objects (w ithin a re stric te d redshift range ( z l,z 2 ) ) in absolute m a g n itu d e M , redshift z ,and equivalent w id th E can be w rit­ ten

N 12(M ,z ,E )d z d M d E = ^ w k$ (M )d V d M x (E )d E 0 |J 10 z (23)

and 0 2 has been defined above. T h e m ag n itu d es m an d m u m m ay rep ­ resent a sp ectro p h o to m etric or o th e r m a g n itu d e ( S ch m id t, Schneider,

k where

©k" = 0 [ m lim(Lk, z ,E ) - m (M ,z ,E )] (24)

and G un n 1986a (SSG) use a B m a g n itu d e and th a t choice is ad o p ted here).

As above, th e discrete counts can be w ritten

N 12( M , z , E ) = X ; < 5 ( M - M i , z - Z i , E - E i ) (25) 2|6(2] ,Z2)

and it is assum ed th a t we can w rite

(26)

S u b stitu tin g these expressions into (23) gives

E - E,) = 5 > „ X > j « ( M - Mj)— x (E)©g*. (27)

k J

N ote th a t all th e inform ation ab o u t the selection function is carried by th e H eaviside function 0 m.

T h e next ste p is to in teg rate over all equivalent w idths E:

£ d ( M - M i, z - z 1) = E “’i,S >i«(M -M j) Jx(E)^0E'c!E (28)

i k i —rv»

Zj <=(*1,z 2 ) W

an d assum e, for th e m om ent, th a t we possess enough inform ation to ev alu ate th e H eaviside function. T h a t is, it is necessary to find the m in im u m equivalent w id th ( E min( Lf., 2, M )) for which th e arg u m en t of th e H eaviside fun ction is positive an d thus th e H eaviside function itself is equal to one. T h en we have re in te rp re te d th e H eaviside function in te rm s of lim its on th e in teg ral as in the cases above and included th e effect of th e selection function exactly.

F inally, th e new lim it on th e integral over equivalent w idth is incor­ p o ra te d a n d th e result is in teg rate d over th e redshift interval (21,22):

z2 00 . . 2 > ( M

- M,)

=

- M

i ) I > k / / x ( E ) d E ^ d z (29) ?,\ i,v .2 ) v.j £ ( '£ i 7/_) ) Z 1 w hich is satisfied if / zr2 °? dV < A i = ( £ < * / J x ( E ) d E — d ZJ . (30) Z1 Emln(Li(,z,Mi)

In o rd er to evalu ate th e integral over equivalent w idth it is necessary to know x(-E) an d , given th is, th ere will be som e m axim um value of E in th e observed equivalent w idth d istrib u tio n above which no objects a re observed, so th a t th e integral over x ( E ) can have any value between

0 and 1 (it is a probability d istrib u tio n ). In p ractical term s, we can use existing d a ta to define x ( E )

-T h e m ajor practical challenge is to ev alu ate th e H eaviside function (i.e. th e selection function) in its full com plexity. T h e selection function will differ in som e degree from one survey to a n o th e r in th a t som e term s may be relevant in some circu m stan ce and irrelevan t in an o th er. Since the present d a ta se t includes both slitless a n d a p e rtu re spectroscopy it is necessary to develop m eth o d s ap p ro p ria te for b o th techniques. F o rtun ately, th e groundw ork for th is ty pe of analysis already exists.

S ch m idt, Schneider, and G unn (1986a, SSG) have developed a com ­ plete m e th o d of analysis for their slitless spectroscopic em ission line surveys. T h e obvious factors which m u st be acco un ted for are th e sys­ tem sen sitiv ity and its variation w ith w avelength, which tra n s la te into th e ab ility to d etect an em ission line o bject w ith a given em ission line equivalent w idth (E ) and co n tin u u m lum inosity at a given redshift. T h e con tin u u m m ag n itu d e m c and equivalent w idth E to g e th er d e te r­ m ine th e flux in th e em ission line w hich is th e fu n d am en tal q u a n tity in th e d etectio n process. D etecting a n em ission line is very sim ilar to d e tectin g a stellar image: in principle it is n o t necessary to d e te c t any con tinu um a t all in order to find a qu asar. T h e sensitiv ity of th e survey is defined by th e search m eth o d used and th e m in im u m signal-to-noise ratio required for d etection of an em ission feature.

T h e less obvious factors included in th e analysis by SSG are those which vary w ith position on th e chip, nam ely position d ep en d en t focus and CCD sen sitiv ity (or illu m in atio n ). T hese factors were d ealt w ith

26

by those au th ors by , in effect, chopping each CCD fram e into sections w ith varying sen sitiv ity zero points and defining effective total survey areas for each zero point. T h e ado pted sen sitivity curve (variation with w avelength) was taken to be the sam e for all areas on all fram es and this is a good ap p ro x im atio n if all the CCD fram es were ob tain ed a t th e sam e airm ass un der sim ilar sky conditions.

B ecause th e SSG sp e c tra were ex tra cted based on corresponding o b je ct positions on th e b road ban d d irect fram e of each field, if is necessary for inclusion in th e sam ple th a t an ob ject be b rig h ter than som e thresho ld used in th e direct fram e au to m atic search algorithm . T h is effect is no t m entioned by SSG and is presum ably negligible. SSG also req u ire th a t all em ission line cand id ates have observed equivalent w idths g reater th a n 50

A

for inclusion in th e sam ple. T hen equation (30) becom es

* .. .

K zi W w here

E ' - m a x [ E min(Lk, z,

Mj),

50/(1 + 2)]. (32)

In sum m ary, SSG req u ire th a t can didates be d etected w ith S/N g reater th a n 7 a n d observed E g reater th an 50A and then define a lim iting co n tin u u m m a g n itu d e in th e

AB

system (O ke an d G unn 1983) a t 6100

A

for such a d etectio n of a line w ith E = 50A . T h e variation of th e sen sitiv ity w ith w avelength

A A B

and th e effect of the actual eq uiv alen t w idth are th en included. T h e resu lt is a lim iting m ag n itu d e

for a line of observed equivalent w idth

E,

observed a t w avelength A on th e k Ul area:

AB?im(

A, E obs) =

Lk

+

A A B ( \ )

+ 2.5log(Eoba/bO) (33) where

Lk

and

AAB(X)

are ta b u la te d functions. T hese eq uation s can be rew ritten for a line a t rest w avelength Ao = A /(l + 2) an d rest fram e equivalent w idth

E — Eoba/(

1 +

z):

A B f i J z , E ) =

Lk

+ A A B ( \ 0( l +

z))

+ 2.5log[E(l +

z ) /

50] (34)

an d it is th is eq u atio n which is solved for

Emin( Lk , z , M)

using th e usual equations to tran sfo rm from absolute to ap p are n t m a g n itu d es, ta k in g care w ith th e band pass in which

M

is defined and including th e

K correction. T h is allows a com plete solution to equation (30) or (31)

and gives an ex plicit n um erical estim a te of th e lu m in osity function. N o te th a t this solves th e inverse problem to th a t considered by SSG although it uses all th e sam e observational relationships. SSG calculate (from the above relatio ns, an assum ed equivalent w idth d istrib u tio n and

an assumed luminosity f un c t i o n) th e expected n u m b er of o b jects an d

com pare this q u a n tity w ith the observed num ber. T h e inverse problem which has been solved here, is to calcu late th e lum inosity fun ction from th e observed num ber counts (b u t still w ith an assum ed

E

d istrib u tio n ). T h e advantage of this approach is th a t no assum ed m odels for th e L F are needed.

An im p o rta n t p o in t to note is th a t all of th e equ ation s discussed refer to a single em ission line a n d its corresponding equivalent w idth

d istrib u tio n ; yet a q uasar may be d etected by several emission lines. A ssum e th a t, at som e fixed z, two lines (Line I an d Line 2) are w ithin th e passband. T h e quasar could be detected by Line 1 a t this redshift, b u t only if Line 1 had an equivalent w idth in th e top 25% of the eq u iv ­ alen t w idth d istrib u tio n w hereas th e quasar could be d etected by Line 2 if its line stre n g th were in the to p 75%. In this situ a tio n it would be correct to do the co m p u tatio n for Line 2. Any case w here m ore than one line is d e te c ta b le will reduce to this ty p e of situ atio n . T h e correc­ tion for th e B aldw in effect (th e correlation of lum inosity and equivalent w idth) can be in corpo rated by using, for th e line in q uestion, the eq u iv ­ alen t w id th d istrib u tio n ap p ro p ria te to th e lum inosity of th e p a rtic u la r o b je ct being tre a te d (recall th a t th is process is done one o b je ct a t a tim e).

C onceptually, th e Va (accessible volume) m ethod works in th e fol­ lowing way. T h e fu n d am en tal assum ption is th a t, over som e redshift range ( z i,2 2), th e LF is a function of M only (not of z). O ne d etects an o b je c t w ith p ro p erties m , E , z w hich tra n sla te into intrinsic p ro p er­ ties M an d E b u t we are in terested only in th e lum inosity represented by M . We wish to co m p u te the co n trib u tio n of this individual o b ject to th e lum ino sity function (later th e individual con trib u tio n s of all th e o b jects will be ad d ed up) so th a t w h at we need to know is th e am o u n t of volum e th a t has been surveyed for this ob ject or, in o th e r words, w hat is th e volum e th ro u g h o u t which this p articu la r o b je ct would have been d etecte d by o u r observing technique. If th e lim its of th e technique were defined sim ply by a lim iting m agn itu de, then one would ju s t be­

gin at th e inner edge of th e redshift range (21), move th e ob ject tow ard higher redshift, adding volum es as one m oved along, u ntil some red shift 2 was reached beyond which th e ob ject could no t have been d etecte d . T h en th e accessible volume Va is all th e volum e sum m ed up o u t to this po int and this o b je c t’s co n trib u tio n to th e LF is th e reciprocal of Va. T h is section has developed th e procedure to tre a t em ission line surveys analogously. O ne finds an o b je ct w ith pro p erties M , 2, and E and, beginning a t z\ m oves th e ob ject in redshift (th e ap p are n t m a g n itu d e an d observed equivalent w id th change as this is done) an d asks w hether th e o b ject would have been d etected by th e survey a t th is redshift. If th e answ er is yes th e n th e volum e elem ent a t 2 is added to th e sum of accessible volum e. It is actu ally m ore involved th a n th is because th e answ er to th e d etectio n question depends on equivalent w idth.

For exam ple, it m ight be th e case th a t, a t th is p a rtic u la r red shift 2, th e o b je c t w ould be found if its rest fram e equivalent w id th were g reater th a n , say, 100

A.

If all QSOs have rest fram e equivalent w idths of this line g reater th a n 100

A

then we would have d etecte d all quasars a t this 2 an d we include th e volum e a t th is 2 in th e sum . On th e o th er han d , if no quasars have E g reater th a n 100

A,

th e n we do not include th is volum e in th e sum because no QSOs could have been d etecte d here. T he need to know th e equivalent w idth d istrib u tio n enters a t this po in t. If h alf of all quasars have lines w ith E > 100,

A

th e n we include th e volum e elem ent w ith a weight of 1/2 (or include 1/2 of th e volum e). T h u s we m ove th e o b je ct in redshift an d sum up th e volum e elem ents

30

d istrib u tio n from E min to oo so th a t the weight is betw een 0 and I.

2 .2 .1 S lit le s s S p e c tr o s c o p y

E q u atio n s sim ilar to those of SSG were used to characterize the detectio n process b u t th e lim iting m agnitudes were w ritten

m u m( z , E ) = jG&(Aq(1 + z)) + 2.5log[E(l + *)/50] (35)

w here m is a sp ectro p h o to m etric m ag n itu d e which differs by 0.0 1 m ag from th e A B system . T h e index k refers to a p a rtic u la r frame. Be­

cause th e re were no can did ates for which convincing reel shifts could be d eterm in e d , no space densities were com p uted for th is sam ple.

2 .2 .2 A p e r t u r e S p e c tr o s c o p y

O n e of th e differences betw een th e slitless and a p e rtu re spectroscopy is th a t th e la tte r d a ta ha,ve featu res in th e sky background and the stren g th s of th e features varies from fram e to fram e. In p articu lar, th e [01] A6300 line varies strongly. T h e background noise in a slitless fram e is n early co n stan t over a single sp ectru m w hereas th e a p e rtu re d a ta show huge variations in background noise, varying from a sm ooth region in th e blue to th e very strong [01] A5577 line, superim posed on w hich it is v irtu ally im possible to d etect any em ission line. T he noise p ro p erties of th e sp ectru m m u st be very carefully d eterm in ed and this m u st be done for each fram e. T his m eans effectively th a t each fram e has its own sen sitivity curve, which is a case th a t has already been discussed.

T h e next p ro p erty of th e a p e rtu re spectroscopy which m u st be taken into account is light loss due to finite a p e rtu re size and im precise cen ­

tering of th e o bjects in th e ap ertu res. T here is no way to avoid allowing each a p e rtu re in th e survey to have its own zero p o in t and it is neces­ sary to in co rp o rate th is reality into th e equations developed up to this point.

T h e procedu re used a t th e telescope selected objects on th e basis of ap p a re n t brightness and ap pearance. T h e V selection lim its chosen were 20 < V < 23.25 an d all objects w hich w ere obviously galaxies were rejected. N ote, however, th a t these crite ria selected th e first p riority ob jects only; stars and galaxies were added back in to th e observation list afte r all first p rio rity o bjects were selected if th e re was spare room left on th e chip for th e low p rio rity targ e ts. T h us th e re is a function

f [ V] which describes th e fraction of objects for w hich a p ertu res were

cu t (as a function of m agn itu de).

Each of the ap e rtu re s actu ally cut falls in to one of two classes: th e first class includes th e a p ertu res th a t were placed on o bjects which are stellar as far as can be d eterm ined (th ey are not obviously galaxies), and th e second class consists of ap ertu res centered on o b jects which are clearly galaxies. In th e QSO lum inosity fu n ctio n calculations, it is necessary to consider only th e form er class of ap ertu res. T his can be m ad e clear by n o tin g th a t we could have declined to observe th e galaxy ap e rtu re s and this would have had no effect on th e nu m ber of quasars u ltim a te ly detecteu . On th e o th e r han d , we could have observed any larger n um b er of galaxies th a n we actu ally did and we would no t have found any QSOs. T h e fractio n of quasars d etected is sim ply th e n u m b er of a p ertu res which co n tain o b jects w hich are possibly stellar divided by

th e to ta l n u m b er of o bjects w hich are possibly stellar. T his fraction is a fur ;tio n of ap p are n t m ag n itu d e.

B ecause it is necessary to tr e a t each a p e rtu re individually w ith re­ sp ect to sensitivity, it is also necessary to define an effective area of the sky w hich is surveyed by each ap ertu re. T h e obvious definition is the to ta l a rea of sky divided by th e nu m ber of ap ertu res available b ut it is necessary only to consider those ap ertu res which do not contain obvi­ ous galaxies; th ese ape 'tures survey no sky area for quasars although th e y do survey for galaxies.

In co rpo ration of these m odifications allows us to w rite th e expected d istrib u tio n of objects:

N 12(M ,z ,E )d z d M d E = — 5 > ( M ) d V d M x (E )d E © rkn0 Kf[V (M ,z)] (36) Hap k

w here k identifies a p a rtic u la r a p e rtu re ,(th e sum is over all ap ertu res ra th e r th a n all fram es) u> is th e to ta l survey area, n ap is th e to tal num ber of a p p ro p ria te ap ertu res, an d f[V(M ,z)] is th e fraction of possibly stellar ob jects w ith a given visual m a g n itu d e for which an a p e rtu re has been cu t. N ote th a t 0™ is defined as in equation (24) b u t th a t th e lim iting m a g n itu d e f or each aperture is d eterm in ed from th e sen sitiv ity curve for th e fram e on which th e a p e rtu re is as well as light loss due to finite a p e rtu re size and o bject cen terin g errors. The m in im u m d etectab le equivalent w id th is found from an equation with th e sam e form as (34).

A pplication of th e identical set of steps perform ed in th e previous sections leads to th e following result.

T h e required functions can be easily defined num erically, from d a ta in th e case of th e equivalent w idth d istrib u tio n , and by a com parison of th e nu m ber of o b jects on th e d irect V fram es w ith th e n u m b er of o b je c ts fo>- w hich spectroscopy was ob tained. T h e equivalent w id th d is­ trib u tio n 1 th a t were used for th e analysis in th is p a p e r were em pirical ones o b tain ed by com bining th e resu lts of W ilkes (19S6), C ra m p to n , Cowley, and H artw ick (1989), B aldw in et al. (1989), an d O sm er (1981). In order to define th e lim iting m a g n itu d e curves, a S /N of 6 in th e d etectio n was required. In p ractice we tran sfo rm e d from th e spec- tro p h o to m etric m m a g n itu d e to a B m a g n itu d e (this is th e b and pass in which th e lum inosity function is defined). T h e integrals (one double integ ral for each ap e rtu re ) were done num erically.

C h a p te r 3

O b se r v a tio n s

T h e observations discussed in th is p aper were m ade during th e course of 2 observing runs at th e C anada-F rance-IIaw aii 3.6m tele­ scope. T h e first run took place in O cto b er 1989 and yielded lim ited resu lts due to poor w eather. T h e second run took place in M arch 1990. D uring this ru n th e w eather was excellent and good d a ta was o btained. T h e observing m eth o d s differed in th a t a p e rtu re spectroscopy (using holes ra th e r th a n slits) was done ra th e r th an slitless spectroscopy as was done d u rin g O cto ber 1989.

3.1 O c to b e r 1989 d a ta

T h e in itial set of d a ta was o b ta in ed a t th e C assegrain focus of th e C anada-F rance-IIaw aii telescope using th e U B C /D A O focal re­ ducer w ith a 75 g ro o v e/m m tran sm ission g ratin g blazed a t 5000

A

and a backside-illum in ated R C A charge-coupled device (C C D ). T h e d a ta were o b ta in ed betw een O cto b er 2 /3 an d O cto b er 7 /8 1989. T h e night of O cto b er 2 /3 was com pletely lost to poor w eather. O bservations were m ade d u rin g th e nights of O ctob er 3 /4 , 6 /7 and O ctober 7 /8 despite th e presence of fog, cirru s, lightning, an d high w inds. A to tal of 15 fields w ere observed b u t th e results are of generally poor and uneven quality. T h ree d istin ct, high galactic la titu d e fields were chosen and m u ltip le CCD fram es w ere o b tain ed w ithin each field. T h e app ro xi­ m a te g alactic la titu d e and longitude of th e fields is: field “sa” (th e “s” prefix indicates so u th ern galactic hem isp here), £ = 49°, b = —47°; field

T a b l e 3.1 L o g o f o b s e r v a t i o n s — O c t o b e r 1 0 8 9 D n to F ie ld I n t e g r a t i o n t im e S e e in g R . A . (1 0 5 0 ) D e c . (1 0 5 0 ) C o m m e n t s O c t . 3 / 4 s a l 3 X 450s 0 .8 5 2 2 :0 2 :0 1 .2 -9 :1 5 :0 1 O c t . 3 / 4 sa2 3 x 450s 0 .9 5 2 2 :0 1 :5 0 .4 - 9 :2 0 :0 0 O c t . 3 / 4 s b l 3 X 450s 0 .0 5 0 0 :3 0 :0 0 .3 0 1 :2 5 :0 0 O c t . 3 / 4 s b 2 2 X 900s 1.0 0 0 :3 0 :0 1 .0 0 1 :2 0 :0 1 O c t . 3 / 4 s b 3 2 x 9 r la 1.1 0 0 :3 0 :0 0 .4 0 1 :1 5 :0 2 s o m e c lo u d s O c t . 3 / 4 s b 4 2 X 900s 1 .0 0 0 :2 9 :5 9 .3 0 1 :1 0 :0 4 O c t . 3 / 4 s c l 2 X 900s 0 .0 5 0 2 :2 9 :5 0 .3 - 4 :1 5 :1 1 c i r r u s O c t . 3 / 4 sc2 2 X 900s 1.1 0 2 :2 9 :5 9 .5 -4 :1 9 :4 0 O c t . 3 / 4 sc3 3 x 900s 1.1 0 2 :2 9 :5 7 .7 -4 :2 4 :3 2 c i r r u s O c t . 0 / 7 s b 5 900 + 343s 1 .4 0 0 :3 0 :0 0 .7 0 1 :0 5 :3 5 a b o r t e d — fo g O c t . 7 / 8 su3 3 X 900s 1.3 2 2 :0 2 :0 2 .0 -9 :2 5 :3 4 fo g a t e n d O c t . 7 / 8 BbS 2 x 900s 1.4 0 0 :3 0 :0 0 .7 0 1 :0 5 :3 5 fo g O c t . 7 / 8 sc4 2 X 900s 1.2 0 2 :2 9 :5 9 .7 - 4 :2 9 :0 9 O c t . 7 / 8 sc5 2 X 900s 1.2 0 2 :2 9 :5 8 .3 - 4 :3 3 :5 5 O c t . 7 / 8 scO 2 X 900s 1.5 0 2 :2 9 :5 8 .7 - 4 :3 8 :3 5 w i n d s h a k e “s b V = 113°, 6 = —61°; field “sc” , £ = 174° , b - - 5 7 ° . T able 3.1 contains a record o f the spectroscopic observations.

A liquid copper sulfate filter th at tran sm itted light in th e ultravi­ o let and blue but cutoff w avelengths longer than 5500 A was used to

obtain the spectra and th e direct (undispersed) fram es. T h e reason this filter was used was to m aintain a sky background that was as dark as possible, because the use of slitless m ode m eans that each point in th e sky background is a m ixtu re of all colors o f sky light. T h e darkest portion of th e sky spectrum is from r<*J3500 - 5500 A (th e region where

36

L ym an a is observed for quasars in the redshift range 1.9 < z < 3.5) and th e sky brightens su b stan tially beyond the brig ht sky em ission line at 5577

A.

T h e p la te scale of th e in stru m e n t was 0.33 arcseconds p er pixel and th e s p e c tra prod uced h ad a dispersion of 18.6 angstrom s per pixel in th e first ord er and 9.6 angstrom s per pixel in th e second ord er (both orders were e x tra c te d ).

A n u m b er of p h o to m etric and sp ectro p h o to m etric sta n d a rd s were observed a t various tim es, altho ug h th e w eather conditions were highly variable so th a t th e calib ratio n of th e d a ta is uncertain. Some of th e fields were observed tw ice in d irec t m ode in order to try to p er­ form a reliable calib ratio n because th e in itial observations were in non­ p h o to m e tric conditions.

3.2 M a rch 1990 d a ta

T h e second set of observations were obtained M arch 26-30 1990 at th e C anada-France-H aw aii Telescope (C F H T ). T h e w eather was clear an d a p p a re n tly p h o to m etric on all four nights. T h e in stru m e n t used was th e C F H T focal red ucer an d PU M A (P U nching M A chine) with th e RCA4 CCD. T h e focal red ucer resulted in an F /2 .9 beam and a scale of 0.30 seconds of arc per pixel on th e 640 x 1024 CCD or a field of view of 16 square arcm inu tes (225 CCD fields equals 1 square degree). A 75 groove per m illim eter grism was used which gave a dispersion of 14

A

p e r pixel or a resolution of ab o u t 45

A.

T h e second ord er was also e x tra c te d and covered th e w avelength range 4000-5000

A

with a

T a b le 3.2 L og o f o b s e r v a t i o n s — M a r c h 1 9 0 0 D a t e F i e ld I n t e g . t im e ( s e c ) S e e in g " A i r m a s s E .A . (1 9 5 0 ) D e c . (1 9 5 0 ) M a r c h 2 0 /2 7 b l 1800 1.1 1 .5 5 1 3 :0 7 :3 0 .1 3 0 :2 0 :0 0 M a r c h 2 7 /2 8 a l 3600 1.0 1.11 1 0 :5 8 :3 0 .2 3 0 :1 0 :0 2 M a r c h 2 7 /2 8 a 2 3600 0 .0 1 .0 4 1 0 :5 8 :4 4 .7 3 0 :1 5 :5 8 M a r c h 2 7 /2 8 b8 3600 1.2 1 .0 3 1 3 :0 7 :3 0 .0 3 0 :3 3 :1 4 M a r c h 2 7 /2 8 bO 3600 1.2 1 .2 2 1 3 :0 7 :4 4 .8 3 0 :3 3 :1 4 M a r c h 2 8 /2 0 u3 3600 1.2 1 .1 8 1 0 :5 8 :5 9 .7 3 0 :1 6 :0 1 M a r c h 2 8 / 2 0 a 4 3600 1.2 1 .0 3 1 0 :5 9 :1 4 .5 3 0 :1 6 :0 0 M a r c h 2 8 /2 0 b l 5 3600 1.1 1 .0 0 1 3 :0 7 :3 0 .0 3 0 :3 7 :3 0 M a r c h 2 8 /2 9 b lO 3600 1.3 1 .0 2 1 3 :0 7 :4 5 .0 3 0 :3 7 :2 9 M a r c h 2 8 /2 0 b lO 3600 1 .4 5 1 .1 0 1 3 :0 7 :5 0 .6 3 0 :3 3 :1 4 M a r c h 2 8 /2 9 c l 5-100 1 .4 5 1.21 1 4 :0 1 :0 0 .0 3 0 :0 9 :5 8 M a r c h 2 0 / 3 0 a 5 3600 1.2 1 .2 3 1 0 :5 9 :2 9 .0 3 0 :1 6 :0 1 M a r c h 2 0 / 3 0 aO 3600 1.1 1 .0 6 1 0 :5 9 :4 4 .0 3 0 :1 0 :0 0 M a r c h 2 9 / 3 0 b l 7 3600 1.1 1 .1 5 1 3 :0 7 :5 9 .6 3 0 :3 7 :2 9 M a r c h 2 0 / 3 0 b l l 3600 1.1 1 .0 3 1 3 :0 8 :1 4 .5 3 0 :3 3 :1 5 M a r c h 2 0 /3 0 b 2 2 X 3600 1.1 1 .0 6 1 3 :0 7 :4 4 .0 3 0 :2 8 :5 9

resolution of 20 A. T h ree d istin ct fields w ere observed w ith galactic la titu d e s and longitudes: field “a ” , I = 200°, b = 66°; field “b ” , I = 72°, b = 85°; field V \ £ = 47°, b = 74°; T able 3.2 is a record of th e spectroscopic observations.

T h e in stru m e n t/C C D com bination (P U M A /R C A 4 ) was chosen for several reasons. It was necessary to cover as large an are a of sky as pos­ sible an d this led us to use a focal reducer ra th e r th a n th e p rim e focus. P U M A is a hole punching m achine for m u lti-a p e rtu re spectroscopy and was available w ith a focal reducer and this led us to ab an d o n th e

slit-38

less spectroscopic m ode which we used for th e O cto b er 1989 run. We again used very low resolution because— although th e slcy brightness was red uced by th e use of ap ertu res— this m inim ized th e significant, read-noise con trib u tio n of the RCA4 chip which was th e best choice of CCD because of its large size and good blue response. In co n tra st to th e U B C /D A O in stru m e n t, the C F H T focal reducer gave good im ages over th e en tire field.

It h a d been suggested (by C hris P ritc h e t and G uy M onnet) th a t we try to use m u lti-a p ertu res ra th e r th a n th e slitless m ethod. We gave co nsiderable th o u g h t to this before deciding in favor of th e m ulti­ a p e rtu re ap pro ach . O u r initial concerns were th e num ber of objects p er field and th e set-up tim e involved w ith aligning ap ertu res with th e objects in th e sky. In th e end th e d o m in an t consideration was th a t one could cover a m uch larger w avelength range w ith ap ertu res than w ith o u t th e m becau se a large bandpass w ith ou t ap ertu res would result in an u n a c c e p ta b ly b rig h t sky background.

W h en one is using m u ltip le ap ertu res in a survey situ a tio n , there .s an o p tim u m n u m b e r of ob jects of in terest per field. W h at this num ber tu rn s o u t to be dep en ds on several factors. O ne factor is sim ply w hether all th e o b jects of in terest can be fit on th e CCD fram e: this depends on th e d ispersion and th e slit or hole size. T h is factor is essentially th e sam e for slitless or m u lti-a p e rtu re situ a tio n s b u t th e re is a small difference: some overlap can be to lerate d in th e slitless m ode if one is searching for em ission line objects. In this case, the sky background is flat an d sm o o th and is easily su b tra c te d . W hen ap ertu res are used,