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Number Counts of Bright Extremely Red Objects Evolved Massive Galaxies at z~1

a r X i v :a s t r o -p h /0404129v 1 6 A p r 2004

Astronomy &Astrophysics manuscript no.ERO˙surfd˙mth.rev2February 2,2008

(DOI:will be inserted by hand later)

Number Counts of Bright Extremely Red Objects:Evolved

Massive Galaxies at z ～1

P.V¨a is¨a nen 1,2and P.H.Johansson 3,4

1European Southern Observatory,Casilla 19001,Santiago,Chile

2Departamento de Astronom ′ia,Universidad de Chile,Casilla 36-D,Santiago,Chile 3Institute of Astronomy,Madingley Road,Cambridge,CB30HA,UK 4

Observatory,P.O.Box 14,FIN-00014University of Helsinki,Finland

Received /Accepted

Abstract.We present results on number counts of Extremely Red Objects (EROs)in a 2850arcmin 2near-infrared survey performed in European Large Area ISO Survey (ELAIS)?elds at K <17.5.Counts of EROs are extended to brighter levels than available previously,giving 0.002±0.001arcmin ?2at K <16.5and consistent numbers with literature values at fainter magnitudes.Photometric redshifts from HYPERZ as well as GRASIL model SEDs of galaxies imply that our EROs are located in the range z =0.7?1.5,with the bulk of the population at z ～1.Taking advantage of the ISO data in the ?elds,we use mid-IR detections to constrain the number of dusty EROs,and also discuss the superior capabilities of Spitzer Space Telescope to detect dusty EROs.Both the mid-IR data and the use of colour-colour diagrammes indicate that at most 10-20%of the EROs in this bright regime are dusty starbursting systems.The space density of our EROs,interpreted to be counterparts of local >2?3L ?massive galaxies at around z ～1,is estimated to be ≈2×10?5Mpc ?3,which is consistent with local values.Furthermore,the cumulative number counts at our bright magnitudes are remarkably well ?tted by pure luminosity evolution models.

Key words.Galaxies:evolution –Galaxies:formation –Infrared:galaxies –Cosmology:observations –Galaxies:starburst –Galaxies:elliptical and lenticular

1.Introduction

Extremely Red Objects (EROs,selected for example by R ?K >5,I ?K >4colours)have received much focus recently by virtue of their potential as a powerful window into the galaxy formation era.By and large,the majority of EROs have been incorporated into a bimodal popula-tion,where the extreme red colours are attributed either to old passively evolving distant (z >1)elliptical galaxies or to extremely dust reddened starburst galaxies (see e.g.recent papers by Cimatti et al.2003,Wold et al.2003,Yan &Thompson 2003,Takata et al.2003,Daddi et al.2002,Smail et al.2002,Roche et al.2002,and references therein for earlier pioneering ERO surveys).It is the class of aged early type galaxies whose number densities provide the strongest constraints on models of galaxy evolution.On the other hand,the dusty EROs could be related to the (ultra)luminous IR-galaxies producing the bulk of the to-tal energy in the Universe since the recombination era (see e.g.Elbaz &Cesarsky 2003for a review).

2V¨a is¨a nen&Johansson:Bright EROs and massive galaxies at z～1

unidenti?ed.The mean redshift was found to be z～1.0 for both populations.Yan,Thompson,&Soifer(2004)on the other hand?nd a small10-15%fraction of isolated passive systems.Yan&Thompson(2003),Cimatti et al. (2003),Moustakas et al.(2004),and Gilbank et al.(2003) have used HST morphologies to di?erentiate between the classes:the results have further complicated the picture, as a large fraction(25-65%)of EROs seem to be disks (though see also Moriondo et al.2000).The expectation originally had been a clearer distinction into spheroids on one side,and irregular and interacting types on the other,expected for extremely dusty star forming galaxies. The relative fractions of di?erent types of EROs thus still remain uncertain(due to for example di?erent selection criteria used)and the results of testing surface densities against galaxy formation scenarios are inconclusive.

In addition to spectroscopic and morphological meth-ods mentioned above,one may separate the old elliptical and dusty EROs by means of longer wavelength observa-tions:any ERO detected at mid-IR to radio wavelengths should belong to the dusty population.Systematically this has been attempted by Mohan et al.(2002)in the sub-mm to radio and Smail et al.(2002)in the radio.Similar fractions(albeit with wide spread)as referred to above of dusty EROs were found in the latter study,while Mohan et al.(2002)?nd a much lower fraction.

In this paper we attempt to separate EROs,for the ?rst time,based on their mid-IR properties.As with far-IR to radio methods,this is a clear-cut de?nition,since the di?erence of ellipticals and starbursting galaxies is very large-distant evolved ellipticals would not be detected with ISO whereas dusty EROs should have strong mid-IR ?ux.Also,mid-IR avoids the identi?cation problems of e.g. large sub-mm beams.We have surveyed an ISO/ELAIS (European Large Area ISO Survey;Oliver et al.2000)?eld in the near-IR and matched the results with an optical dataset.We make use of the newly available Final ELAIS Catalogue(Rowan-Robinson et al.2004).In this paper we consider the surface densities of EROs resulting from a wide-?eld relatively shallow J and K-band survey.We shall present the results from a deeper NIR survey around faint mid-IR detections in subsequent papers(V¨a is¨a nen& Johansson2004;Johansson,V¨a is¨a nen&Vaccari2004,in prep.).

The structure of this paper is as follows.The observa-tional data are presented in§2,and in§3we de?ne and extract the ERO sample from the photometric catalogues. In§4we discuss the dusty vs.evolved ERO separation and in§5the SEDs of the detected EROs.The resulting sur-face densities of EROs are presented in§6.We assume throughout this paper a?at(?0=1)cosmology with ?m=0.3,?Λ=0.7and H0=70kms?1Mpc?1.2.Observations

2.1.Near-Infrared data

The observations presented here were conducted in the ELAIS?elds N1and N2,centered at(α,δ)= 16h09m00s,54?40′00′′and(α,δ)=16h36m00s,41?06′00′′J2000.0respectively.

The near-IR survey performed with the Mt.Hopkins 1.2-m telescope,reaching limiting magnitudes of approx-imately J=19.3and K=17.5,is fully presented in V¨a is¨a nen et al.(2000;hereafter V00).The data cover a total of one square degree,although a slightly smaller area is used here(2850arcmin2)after some edges and higher noise areas were excluded and,more importantly,only ar-eas with full coverage in both bands were considered.The detector used was a256×256InSb array with1.2′′pixels.

2.2.Optical data

We obtained optical photometric data from the Isaac Newton Telescope(INT)Wide Field Survey(WFS). The WFS data are publicly available on the Cambridge Astronomical Survey Unit(CASU)homepage1.For a re-view on the INT Wide Field Survey Project and instru-ment characteristics,see McMahon et al.(2001).The data consist of U,g′,r′,i′,Z band photometry(Vega-based)in the N1region and g′,r′,i′,Z photometry in the N2re-gion(the apostrophes are dropped henceforward for clar-ity).The WFS bandpasses are similar to the SDSS(Sloan Digital Sky Survey)?lters(Fukugita et al.1996).The WFS webpage2gives colour transformations between the WFS?lters and those of the standard Johnson-Cousins system(Landolt1992):r?R=0.275(R?I)+0.008and i?I=+0.211(R?I)for the bands most used in this paper.The nominal5σdetection limits for a1′′seeing are g≈25.0,r≈24.1,i≈23.2,and Z≈22.0(the WFS page).For further details on the pipeline processing of INT wide?eld survey data consult Irwin&Lewis(2001), and Gonzalez-Solares et al.(2004)for more in-depth dis-cussion of the WFS data in particular in ELAIS?elds.We also obtained raw R-band images of the ELAIS N1and N2 regions from the archives.After reduction we used them for object identi?cation–only the pipeline processed cat-alogues were used for photometry however.

2.3.Astrometry and photometry

The INT astrometry was derived using Guide Star Catalog (GSC)stars,which results in an external astrometric ac-curacy of0.5′′?1.0′′(Irwin&Lewis2001).Astrometric ac-curacy in the Mt.Hopkins data ranges between0.5?1.5′′, and was calibrated using GSC and US Naval Observatory (USNO)catalogues.

All our near-IR photometry is performed using the SExtractor software(v.2.2.1and v.2.3;Bertin&Arnouts

V¨a is¨a nen&Johansson:Bright EROs and massive galaxies at z～13 1996).Our NIR photometry is explained in V00.As dis-

cussed therein,we used total magnitudes as given by the

SExtractor‘BEST’-magnitude.However,since the optical

WFS photometry was given only in2.4′′diameter aper-

ture magnitudes,we re-did all the old photometry with

matching apertures.It is crucial that the di?erent obser-

vations map the same region of the source when construct-

ing colour indices;near-IR magnitudes not associated with

colours are given as total magnitudes in this paper.

The archive optical INT WFS data in the ELAIS re-

gions come as fully calibrated photometric source cata-

logues(Irwin&Lewis2001;Gonzalez-Solares et al.2004).

No further processing is done for the catalogue except

merging of multiple and/or nearby sources,as described

below in Section3.2.

3.Construction of the ERO samples

3.1.De?nition of EROs

Numerous di?erent selection criteria have been de?ned for

EROs,including R?K≥6,R?K≥5.3,R?K≥5,

I?K≥4with K-magnitude upper limits from18to

21mag.All these criteria are designed for selecting early

type galaxies at z≥1.In this paper we use the following

de?nition for EROs:r?K≥5.5and/or i?K≥4.4.Our

limits result from colour transformations given above in

Section2.2for typical colours of R?I≈1.4?2.0of our

sources,and from the desire to have limits corresponding

to the commonly used R?K>5,I?K>4selections

to aid comparisons to other surveys.In addition,we will

check any results with r?K>5.8,i.e.R?K≥5.3,which

is also often used.

Furthermore,we calculated r?R and i?I colours us-

ing SEDs from the GRASIL library(Silva et al.1998)and

veri?ed the r?R≈0.5and i?I≈0.4colours to match

well those of ellipticals at z～1.Naturally an exact com-

parison or transformation between di?erent galaxy colour

selection criteria is redshift and galaxy SED dependent,

and we note that since the model colours that di?erent

authors use vary,there might easily be0.1–0.3mag di?er-

ences in the colours at z～1.Elliptical galaxies become

EROs at z≈1.1?1.2with the criteria and models we

used.

Fig.1shows r?K and i?K model colours of several

representative galaxies against redshift with the ERO cri-

teria included.Model SEDs are adopted from the GRASIL

library(Silva et al.1998;model SEDs and the GRASIL

code are available online3).Ordinary spirals(dotted line)

never reach the red colours of EROs,while both ellipticals

(solid)and reddened starbursts(dash-dot)become EROs

when seen beyond z～1.For comparison,the colour of the

prototype dusty ERO HR10is also plotted(dashed curve)

as a function of redshift.The colour of the HR10model

is due to extreme dust extinction(Silva1999).The lowest

panel shows the?ux ratio f15μm/f2.2μm–the degeneracy

4V¨a is¨a nen&Johansson:Bright EROs and massive galaxies at z～1

some remaining NIR frame cosmic rays,etc.,and though a handful of these may be genuine very red objects(though not necessarily EROs)we decided to conservatively ex-clude all of these from the?nal source list.

Since the WFS catalogue contains multiple detections of the same source(due to overlapping CCD frames),opti-cal counterparts within0.5′′of each other were averaged. After this purging,the remaining multiple matches within 1.5′′were summed up since the corresponding NIR cata-logue would not have resolved them as individual sources, and then the brightest source was selected if there still were multiple counterparts available.Approximately18% of the original WFS optical catalogue closest matches were a?ected by the purging,though less than4%by more than 0.1magnitudes.

3.3.Star vs.galaxy separation

We then proceeded to separating stars from galaxies in the surveys.Stellarity indices were available from both the NIR catalogue(the SExtractor CLASS parameter)and the WFS catalogue,where a?ag de?nes galaxies,de?nite stars,and various degrees of uncertain stellarities.The WFS classi?cation is in principle more useful for us here, since it goes deeper than the NIR data.Note that the star-galaxy separation is di?erent from that conducted in V00since we now have a much deeper optical catalogue available.

However,for the EROs we are interested in,the red-dest and faintest objects in the catalogue,we?nd that a colour separation works best.This can be seen in Fig.2, which shows the full catalogue in r?i vs.r?K with the morphological classi?cation indicated:stars are over-plotted as small crosses in the left panel.A separating line,adjusted experimentally to maximally distinguish the main concentrations of stellar and extended sources in the ?gure,of r?K=2.16(r?i)+1.35is drawn.This colour-colour diagramme separates stars very well from galaxies; it is important to note that there is virtually no overlap between the stellar sources and galaxies in the region of interest at r?K>5.The r?i colour,in fact,is very important in the separation of red stars.Very low mass and cool stars(of the L spectral type in particular)have R?K and J?K colours closely mimicking those of extra-galactic EROs–for these stars,however,R?I(or r?i) always stays above≈2(see e.g.Chabrier et al.2000;Cruz et al.2003).On the other hand,we?nd no galaxy models resulting in r?i>2colours(GRASIL code used;see also eg.Fugukita et al.1995).All extremely red objects with r?i>2can safely be discarded as stars.

Representative galaxy colours calculated from GRASIL models(Silva et al.1998)are overplotted in the right panel.In fact,the elliptical model does overlap with the r?i～3extremely red stars,but only at z>4. This is a potential concern with deeper ERO surveys,but at our brighter magnitudes such distant sources are not expected to be seen.

We thus use a combination of methods.We de?ne as stellar all objects having stellar colours according to the above limit and those not having a galaxy morphology set by the WFS survey.The end result of this is that brighter sources are preferentially separated by morphology and fainter ones(especially red sources)by colour.The star counts were already compared to the SKY model predic-tions of Cohen(1994)in V00,and found to?t well model predictions for the corresponding?elds.In the regime K<17.5,45%of r?K>5.5sources were classi?ed as stars.In summary,it should be stressed that in the case of bright surveys of EROs,the contamination from red late type(typically L-type)stars is considerable when using the R?K based selections of EROs.

3.4.Extraction of EROs

We searched for EROs from the Mt.Hopkins galaxy cat-alogue according to the colour de?nitions given above in Section3.1.Only NIR detections at5σlevel and over were considered.We then went through the resulting list check-ing our NIR maps as well as the WFS CCD images for any obvious spurious objects(for example,anomalously low optical magnitudes were found in cases when sources fell close to gaps in the CCD frames,or near bright stars –these were excluded).Ultimately there were50EROs in the matched Mt.Hopkins catalogue using the r?K crite-rion and21using the i?K limit-17EROs are common to lists resulting from both selection criteria.This makes 54EROs in total.All the EROs have an i-band detection, while there were4EROs which have only upper limits in the r-band.It is obvious that the r and i band based selection criteria for EROs are not equivalent.There are more than twice as much r?K>5.5selected EROs than i?K>4.4selected ones.

The photometry for all EROs is given in Table1and the resulting total numbers of veri?ed EROs are summa-rized in Table2,along with some other survey character-istics.The colour-magnitude plot is shown in Fig.3.

We wish to point out,that in order to be very con-servative in limiting the number of spurious detections in ?nal ERO lists,as well as to be able to take advantage of the J?K colour in classi?ying EROs,we required both a J and a K detection for all considered objects from the survey.While the5σdetection requirement was applied only to the K-band,this nevertheless might exclude some genuine EROs which are very red in J?K colour.At our faintest levels of K≥17.5,we do not expect to detect any J?K≥2.5EROs,and already at K≥17the complete-ness of J?K≥2EROs would not be quite as high as J?K<2EROs.

4.Separation of dusty EROs

Since the Extremely Red Objects can be divided into two broad classes,the populations should be di?erentiated be-fore surface density comparisons to detailed galaxy for-mation models are attempted.A prime motivation of this

V¨a is¨a nen&Johansson:Bright EROs and massive galaxies at z～1

5 Fig.2.Observed r?i vs.r?K colours of our survey.Stellar sources,as de?ned by the WFS classi?cation are overplotted as small crosses in the left panel.It is seen that this colour-colour diagramme does a very good job in separating stars from galaxies:in particular there is virtually no overlap in the region of interest,at r?K>5.Any “ERO”with r?i>2is classi?ed as a star.The right panel shows all the galaxy EROs of the Mt.Hopkins survey selected with r?K>5.5and several GRASIL model SEDs overplotted:the lowest solid curve is the SED of an Sb galaxy,the dashed curve reaching ERO regime is that of a starburst,and the highest curve an elliptical.The curves are plotted up to redshift of z=2(see text and Sect.3.1).

study is to attempt the separation using the mid-IR data. We shall also perform the separation using other estab-lished methods,and discuss the di?erences.

https://www.sodocs.net/doc/21581728.html,ing mid-IR data

Since only dusty EROs are expected to show up in the mid-IR ELAIS survey,the question to answer is,what fraction of the EROs are detected in the mid-IR to a given ?ux limit?

Among the54Mt.Hopkins EROs we?nd only one match with a15μm ISOCAM source.In the ELAIS band-merged catalogue(Rowan-Robinson et al.2004)the source is a candidate hyper luminous IR-galaxy based on a pho-tometric redshift of z≈1.0,which agrees with our inde-pendent photometric redshift determination of z=0.9. (We note that using the same near-IR data-set but with only K-band considered,another ISOCAM detected ERO is found in Rowan-Robinson et al.(2004)).Thus,at face value,the fraction of dusty EROs seems to be insignicant.

However,it is crucial to take into account detection limits.The main hindrance to the full power of the mid-IR separation with our data-set is the relative shallow-ness of the ELAIS survey.All the Mt.Hopkins EROs have K=16?https://www.sodocs.net/doc/21581728.html,paring to GRASIL models(Fig.1), a starburst galaxy with M82-type SED and K=17ap-parent magnitude should just have been detected over the 0.7mJy ISOCAM?ux limit.In comparison,to detect the Rayleigh-Jeans tail of ellipticals would have required approximately K=13brightness.We note that M82-type galaxy with K=17mag at z=0.7and z=1.0 translates to star formation rates SF R≈100and200 M⊙yr?1,respectively,as calculated from total IR lumi-nosity L IR(3?1000μm)with relation given e.g.in Mann et al.(2002)and using the GRASIL model SED.The cor-responding IR luminosities are log(L IR)>11.7.

There are23K<17EROs in the sample,out of which only one(4%)is detected in the mid-IR.As shown above, the ELAIS data is only sensitive to luminous IR galaxies at the expected redshifts of EROs(z>0.7).Thus,we can state that very strong starbursts(SF R>100M⊙yr?1 make up only a small fraction,less than10%,of counter-parts to bright EROs.For detection of more modest dusty galaxies other methods have to be used.

It is relevant to note that the in-orbit Spitzer mission will de?nitely?nd large numbers of dusty EROs,including more modest ones than above:Using the M82SED once again as an example,sampling dusty K≈20EROs at z≈1would mean probing dusty starforming galaxies of SF R～10M⊙yr?1.The expected?ux densities of such objects with IRAC8μm and MIPS24μm bands would be of the order of10and150μJy,respectively,which are easily reached in a few minutes of integration time. For example,assuming10minute per pixel integrations, a10hour survey to the mentioned5σdepths would cover ～800arcmin2in both bands.This means detecting100–

6V¨a is¨a nen&Johansson:Bright EROs and massive galaxies at z～

Fig.3.The K-band magnitude vs.the r?K and i?K colours,in left and right panels,respectively.All objects coming through the catalogue construction as galaxies are plotted as open circles.Individually veri?ed EROs at>5σlevel are overlaid with solid circles.Since the survey was done in ELAIS?elds,we also overplot all matched ISOCAM detections as solid triangles.Note that the total K-magnitude is used on the x-axis,whereas the colour is calculated with matching small apertures.

200MIR counterparts of dusty EROs,assuming average

ERO surface densities and an ad hoc50%fraction of dusty

EROs.

4.2.Colour-colour selections

Figure4shows the colour-colour separation scheme of

EROs by Pozzetti&Mannucci(2000)adopted for our r,i

?lters.The idea is that the J?K separates the EROs at

z>1to bluer early type galaxies(where the large optical

to near-IR colour is due to the4000?A break),and to the

redder(in J?K)dusty EROs which have a a smoother

SED from optical to near-IR.In the r?K plot we?nd

9/50r?K selected EROs in the dusty starburst side of

the indicator,and similarly3/16of the i?K EROs.In

the i?K plot many r?K EROs fall under the i?K=4.4

line,and the dusty percentages are higher for the remain-

ing EROs,30?50%.The one mid-IR ERO is overplotted

in the?gure,and it in fact falls on the elliptical side of

the division.However,given the typical photometric er-

rors of the EROs(the error bars on the mid-IR ERO are

representative)it is seen that at least1/3of the EROs fall

statistically on the dividing line.Moreover,while the ellip-

tical vs.starburst separating line is equivalent to Pozzetti

&Mannucci,it is only de?ned to work for R?K>5.3.If

we thus select EROs at r?K>5.8,nearly all galaxies on

the starburst side of the left panel in Fig.4fall out,and

the elliptical fraction becomes close to90%.Finally,our

Mt.Hopkins sample is1-2magnitudes brighter than other

typical ERO samples,and it is therefore not certain how

accurate the separation should be.

We also checked the colour-colour separation scheme

presented by Bergstr¨o m&Wiklind(2004)using R?J

vs.J?K colours(see their Fig.8).The result is85%

ellipticals regardless of whether r?K or i?K ERO crite-

rion(or both)is used,totally consistent with the fractions

emerging from the Pozzetti&Mannucci R?K vs.J?K

method.

We thus conclude that using the various J?K vs.

optical-NIR colour separation schemes approximately80%

of our EROs appear ellipticals.Only when using Pozzetti

&Mannucci i?K based separation for i?K>4.4EROs,

is the percentage lower,≈60%.We might be missing the

very reddest J?K EROs,i.e.some dusty EROs,because

of the selection of the sample in both bands.At K<17,

where there should be no bias in J?K colour,the fractions

of early type EROs range between65-80%,depending on

the optical band used.

5.Photometric redshifts and SEDs

We calculated photometric redshifts for the EROs using

the HYPERZ code(2000)which?ts GISSEL98synthetic

spectra(Bruzual&Charlot1993;BC hereafter)galaxy

templates to photometric data points.Figure5shows

the photometric redshift distributions of our sample.The

mean photometric redshift is z=0.94±0.49,excluding

the outliers at z～4.However,it is clear that subgroups

are involved,though the gap at z=1?1.2is di?cult to

understand.We also note that approximately half of the

V¨a is¨a nen&Johansson:Bright EROs and massive galaxies at z～1

7 Fig.4.J?K vs.r?K and i?K ERO discriminator colours adopted from Pozzetti&Mannucci

(2000).All galaxies are plotted as dots and EROs are overlaid with larger symbols.Squares are r?K>5.5EROs and asterisks i?K>4.4 EROs.Those i?K EROs with no r-band detection have lower limits indicated in the left panel.The only mid-IR detected ERO is shown as the large solid symbol–its error bars are representative to errors of all EROs.

?ts result in a con?dence better than>85%,so the results

should be taken with some caution.On the other hand,

the overall distribution,average redshifts,or properties of

sub-groups discussed below do not change if we consider

only those EROs with?ts of>85%con?dence.

The21i?K selected EROs,the white regions in the

histogram,are more evenly distributed and at higher z

(average z≈1.4)than r?K EROs.The17objects which

are EROs with both selection criteria cover quite evenly

the range z=0.6?1.8(average at z≈1.3).The EROs

#51–54in Table1are all between z=1.4?1.9.

Most notably,however,those EROs that are not EROs

with the i?K>4.4criteria,show a signi?cantly nar-

rower redshift distribution(hatched region of histrogram)

with z=0.73±0.06.Moreover,these objects constitute

more than half,33/54,of our total ERO sample–we re-

turn to these below in Section6.2.Taken together,the

photometric redshifts thus suggest that r?K>5.5se-

lects more nearby systems at redshift unity and below,

and i?K>4.4includes a wider range of EROs.

Virtually all,50/54,of the best-?t HYPERZ BC SEDs

are either starburst spectra of age0.4to2Gyr,or ellipti-

cal spectra at several Gyr–these SEDs are nearly iden-

tical(see Fig.6).In fact,in a recent paper by Pierini et

al.(2004)it was proposed that post-starburst ellipticals,

forming in a short bursts during the period where EROs

are observed(between1

stituent of ERO populations.It should be stressed that ba-

sically all of the best-?t HYPERZ’starbursts’are evolved

results of instantenious bursts and not dusty starforming

galaxies.

Fig.5.Photometric redshifts as calculated by HYPERZ.

All54Mt.Hopkins EROs are plotted,whatever the con?-

dence of the?t(half have>85%).r?K>5.5selected

EROs which are not EROs with i?K>4.4selection are

hatched,the empty regions of the histogram thus show

the i?K>4.4EROs.

8V¨a is¨a nen &Johansson:Bright EROs and massive galaxies at z ～

Fig.6.Averaged and K -band normalized SEDs of i ?K EROs (circles)and those r ?K EROs with i ?K <4.2(squares).The observed bands are g,r,i,Z,J,K .The GRASIL M82model and elliptical and evolved starburst SEDs from BC are also plotted at indicated redshifts.

6.Discussion

6.1.ERO counts and surface densities

The numbers of detected EROs are given in Table 3,and the corresponding cumulative number counts are plotted in Fig.7.The numbers plotted are the total ERO counts:based on the previous discussion,the fraction of dusty EROs is low (～15%)and does not have signi?cant e?ect even if subtracted from the counts.

Our wide survey from Mt.Hopkins extends the ERO number counts to brighter magnitudes than observed be-fore.We report the ?rst ERO counts at K <16.5,where the r ?K >5.5(ie.R ?K >5)selection yields a surface density of 0.002±0.001arcmin 2,and the i ?K >4.4(ie.I ?K >4)selection ≈50%of this value.At K <17mag-nitudes we arrive at 0.007±0.002arcmin 2of r ?K EROs.Beyond K >17mag the Mt.Hopkins survey is starting to be a?ected by incompleteness,and based on simulations performed in V00for the same dataset,we estimate a fac-tor of 1.5correction to the K <17.5cumulative ERO density.The resulting r ?K selected surface density at K <17.5is 0.018±0.003arcmin 2.

We compare these numbers to the only two other ?eld ERO surveys with enough sky coverage to reach these bright magnitude levels:The largest ERO survey to date Daddi et al.(2000a;with 701arcmin 2),yielded 0.003±0.002arcmin 2at K <17and 0.02±0.006arcmin 2at K <17.5.These are totally consistent with our result.The Yan &Thompson (2003;409arcmin 2)HST ERO counts were selected using I ?K >4.0.At K <17they agree with our corresponding i ?K >4.4selection within

the errors.At K <17.5there is a clear discrepancy,how-ever.It is our r ?K selected EROs which are in closer

agreement with the Yan &Thompson ERO counts –we systematically ?nd a factor of ～2less i ?K selected EROs than r ?K selected ones.Especially at the faintest bin there appears to either be more incompleteness than we expect,or the i ?K >4.4selection does not pick up as many bright K ～17galaxies at z ～1than the r ?K selection does.Indeed,the discussion on “excess”r ?K EROs in the next Section 6.2and the photometric red-shifts derived for the EROs hint at the latter possibility.

6.2.Evolved ellipticals at z ～1

The strength,and motivation,of ERO searches has been to look for the most massive galaxies at redshifts of unity and over (see e.g.Saracco et al.2003),since it is exactly these which place tight constraints on galaxy formation scenarios.The dusty vs.early type ERO discriminators showed that most of our ERO sample is consistent with being early type galaxies.However,we argue that the clearest population of ellipticals comes from the class of r ?K >5.5EROs which have r ?K <4.4–these consti-tute 60%of all our EROs.Fig.8shows an i ?K vs.r ?K plot,where these “excess”EROs populate the upper right quadrant (see also Figs.4and 6).

Intuitively,the upper left quadrant galaxies should be sources at a redshift where a signi?cant 4000?A break falls right in between the r and i bands:i.e.ellipticals or very early type spirals at a redshift z ～0.8.We calculated numerous GRASIL models,and it turns out to be fairly di?cult to obtain colours in this ERO regime.Any signi?-cant on-going star-formation drops the r ?K colour down,as noted also by e.g.Yan &Thompson (2003;see their Fig.9.).The e?ect is seen in the solid curve of the right panel of Fig.8:the elliptical model has a formation red-shift of z ?=3and at redshifts of z >1.5the e?ects of the starburst turn the r ?K colour sharply down.Signi?cant amounts of extinction,including edge-on spirals,do not bring models to the r ?K >5.5,i ?K <4.4region either:the dotted line depicts the extremely dusty HR10model as an example.

One gets closest to the r ?K >5.5,i ?K ～4re-gion by merely redshifting an old present-day elliptical to Sa galaxy to the appropriate redshift.This is shown as the dashed curve,a 13Gyr old Sa spiral;any >10Gyr old elliptical produces a similar result.While this is an unphysical model,it serves to point out that while pure reddening is unlikely to reach this part of the colour-colour diagramme,an old stellar population does that.We thus conclude that EROs which are selected by r ?K >5.5(i.e.R ?K >5)which are not EROs by i ?K >4.4(i.e.I ?K >4)are mostly early type galaxies.In fact,at i ?K <4.2there should not be any contamination from dusty galaxies (compare the vertical dotted line to the dusty model curves).

V¨a is¨a nen &Johansson:Bright EROs and massive galaxies at z ～1

Fig.7.Cumulative ERO counts are shown for R ?K >5and R ?K >5.3selections (ie.r ?K >5.5and r ?K >5.8),

in the left and right panels,respectively,with solid triangles.Additionally,the i ?K >4.4selected EROs are plotted in the left panel with circles.Error bars are Poissonian.Our other ?eld ERO counts from a deeper survey (IRTF,V¨a isanen &Johansson 2004)are shown as stars.Squares show the EROs of Daddi et al.(2000a)and the asterisks the Roche et al.(2002)EROs in the left panel.The left panel shows R ?K >5selected EROs of Daddi et al.(2000a)and Roche et al.(2002).In addition to the Daddi et al.R ?K >5.3selected EROs the right panel plots the corresponding Smith et al.(2002)and Smail et al.(2002)EROs.We plot PLE models of Daddi et al.calculated using the indicated R ?K cuts.The curves are di?erent by their galaxy formation redshifts,as indicated in the ?gure.The two dashed-lines employ a di?erent LF –the higher curve results from a 2MASS normalized LF,and the lower from that of Marzke et al.(1994).

We also plot a sample of averaged SEDs of our EROs in Fig.6.The “excess”EROs with i ?K <4.2are plot-ted with squares and are well ?t by very evolved galaxy SEDs,as just discussed.The critical observed band is the i band,where the di?erence between elliptical and dusty starbursts (M82plotted)is the largest at the redshifts of our targets.Note that the g -band is biased to the bluest sources,since the average is calculated from those EROs detected in the corresponding band,and only 25%of the sources have a g -band detection.

The r ?K and i ?K selections alone are quite di?er-ent (see e.g.Scodeggio &Silva 2000).At least with bright ERO surveys such as ours,it seems that r ?K selects more passive systems.This is consistent with spectroscopic fol-lowup of R ?K >5.3surveys ?nding signi?cant amounts of ellipticals (e.g.Cimatti et al.2002)and those following up I ?K >4selections ?nding much more star formation (e.g.Yan et al.2004).

6.2.1.Number densities

There are 33“excess”r ?K EROs with i ?K <4.4.They appear to be at redshifts z =0.7?1.2:BC SED ?tting favours redshifts below unity (Fig.5),while the GRASIL models suggest redshifts at z ≈1?1.2(eg.Fig.8).These EROs have typical magnitudes of K ≈17.2,which would

make them at least ～2L ?galaxies,taking into account passive evolution since that redshift.Very simply,assum-ing that this group of EROs consists of ellipticals and is constrained in the mentioned redshift range we derive a co-moving volume of 2×106Mpc 3for the Mt.Hopkins survey area,and thus a space density of ～2×10?5Mpc ?3.Taking K -band LF parameters from Kochanek et al.(2001),this is a signi?cant part of the expected density of ≈8×10?5Mpc ?3for local >2L ?early type galax-ies.Given the incompleteness in our survey at K >17,and that we did not include ellipticals at i ?K >4.4in this calculation these ERO counts are a very conservative lower limit,and thus indicate that a major part of massive present day ～2?3L ?ellipticals were in place at z ～1.

The space density just derived is intriguingly close to those estimated for the higher redshift sub-mm popula-tion,which are suspected to be the most luminous star-bursts,perhaps the progenitors of present day massive el-lipticals (see e.g.discussion in Scott et al.2002).

6.3.Constraints on galaxy formation scenarios

Figure 7plots our ERO counts along with several other recent surveys and pure luminosity evolution (PLE)mod-els of Daddi et al.(2000a).All curves depict ellipticals with a τ=0.1Gyr initial starburst,and passive evolution

10V¨a is¨a nen&Johansson:Bright EROs and massive galaxies at z～

Fig.8.i?K vs.r?K colour-colour magnitude diagramme of the veri?ed EROs is plotted in the left panel.The mid-IR detected EROs is overplotted as a large solid symbol.Several GRASIL models are plotted in both panels:the models are drawn until redshift z=2,and the z=1locations are indicated by small solid circles along the curves. The reddening direction is indicated in the left panel by the diagonal arrow which correcponds to A V=1extinction. afterwards.The di?erent lines have formation redshifts of

z form=2.4,3.0,and10.All the models use a Marzke et

al.(1994)LF,except the uppermost dashed line,which

instead uses a2MASS based LF(Kochanek et al.2001)

resulting in a higher normalization by approximately a

factor of two.

We stress that all EROs are counted in the?gure,i.e.

no corrections for the dusty population have been made.

As was seen previously,this is perfectly reasonable for our

very bright sample of EROs.However,even a very large

correction(not supported by our observations)of50%

of the EROs being in fact dusty,would not change the

conclusion signi?cantly.At somewhat fainter magnitudes,

recent studies have shown(Cimatti et al.2002,Smail et

al.2002)that the percentage of dusty EROs is likely some-

where around30–60%.

The PLE model using a formation redshift of z form=3

?ts all the data remarkably well.Formation redshifts

slightly lower than this also?t well our bright ERO counts,

but start underpredicting the numbers at fainter magni-

tudes.On the other hand,corrections for dusty galaxies

might be larger at the fainter levels(see eg.Smith et al.

2002).However,models with z form<2.4start underpre-

dicting the counts at all levels,especially when using the

R?K>5.3selection criteria.The very highest formation

redshifts of z form=10and over,predict steeper number

counts and are more di?cult to?t to both the bright and

faint end of ERO counts with the same normalization.The

?gure shows two di?erent LFs giving a factor of～2di?er-

ence.Note also that eventhough the R?K>5.3criterion

eliminates nearly all the(possibly)lower redshift z～0.8

EROs,there is no signi?cant di?erence in the?ts of PLE

models in the right and left panels of Fig.7.

Using the2MASS normalized LF it is in fact slightly

more di?cult to?t both R?K>5and R?K>5.3

counts simultaneously.The R?K>5counts are overpre-

dicted by a factor of1.5?3with all except the very lowest

z form=2.2?2.4formation redshifts.Good?ts in the range

K=16.5?18.5are again acquired with z form≥4,but the

faintest counts are overpredicted.Roche et al.(2002,2003)

?nd their2MASS-LF based PLE models signi?cantly over-

predicting I?K>4ERO counts,and show that certain

amount of merging and density evolution?ts the counts

much better in particular in the fainter K>20regime.

That a range of formation redshifts are necessary to

model EROs is seen for example by comparing Figs.8

and7(see also Cimatti et al.2003):though models with

formation redshifts of z≈3?t well the counts,the colours

produced by such a scenario are not red enough for many

EROs found in this survey,and others.Very red colours

of r?K>6.5or i?K>5can not be produced without

pushing z form closer to10(unless all such extreme EROs

are heavily reddened starbursts).

We do not investigate hierarchical formation scenarios

in more detail here.We merely point out that predictions

from some recent models(Cole et al.2000as presented in

Smith et al.2002),fall short an order of magnitude in the

numbers of EROs in the range K=17?20.See also e.g.

discussion by Martini(2001)and Firth et al.(2002).It is

important to realize that a drastic increase in the fraction

of dusty EROs does not make the hierarchical models?t

the ERO counts any better:they predict too few EROs of

V¨a is¨a nen&Johansson:Bright EROs and massive galaxies at z～111

all kinds.On the other hand large fractions of dusty EROs would have to be subtracted from PLE models including only ellipticals,making all the Daddi et al.PLE models used above to overpredict the counts by factors of1.5?3.

It is concluded that the PLE models do give remark-ably good?ts to the brightest ERO counts.Formation redshifts around z～3are favoured–however,by alter-ing the details of the models,formation redshifts between z=2?10are also consistent.Moreover,a range in the formation era of ellipticals is suggested by the range in ERO colours.

7.Summary

We have searched for EROs in a near-IR survey performed in ELAIS?elds,using r?K>5.5,r?K>5.8,and i?K>4.4colour criteria.These are equivalent to the commonly used R?K>5,R?K>5.3,and I?K>4. In the survey,reaching approximately K=17.5,we?nd 54EROs.The area covered is2850arcmin2.

Taking advantage of overlapping mid-IR data,we search for dusty EROs,since only these should be de-tected with the used15μm ISOCAM band.Only one is found from our conservatively constructed catalogue. Taking into account detection limits we limit the number of very strong starbursts(SF R≥200M⊙year?1)in the bright K<17?17.5ERO population to<10%.

We also make use of a J?K vs.optical-infrared colour-colour diagramme to separate EROs,and?nd that the fraction of dusty ERO population is<10?40%,depend-ing on the colour used.There are more dusty galaxies in the i?K based ERO selection than if r?K is used. HYPERZ photometric redshifts and template?ts are also employed:nearly all redshifts are in the range z=0.6?1.8 with a strong peak at z～0.8.Approximately90%of the best-?t SEDs are those of evolved stellar populations.

We?nd a considerable amount,～60%of all our EROs, of r?K>5.5EROs which are not EROs with the i?K>https://www.sodocs.net/doc/21581728.html,ing models we interpret these to be early type galaxies at redshift of z～0.7?1.1.They are interpreted to be the counterparts of local2?3L?galax-ies,and their resulting space density is approximately 2×10?5Mpc?3.

Cumulative number counts are provided for the EROs, extending the available ERO counts to brighter limits than previously.Our counts are consistent with litera-ture counts in the overlapping magnitude region with same colour cut-o?s.

Our ERO number counts,as well as other literature data,are well?t by pure luminosity evolution models. Formation redshifts for early type galaxies in excess of z=2.5are required to?t the ERO counts,and z≈3 is favoured.However,the range in the colours of EROs suggests also a wide range in formation redshifts. Acknowledgements.We thank the referee for good and clarify-ing comments.We are grateful for useful discussion with Kalevi Mattila,Emanuele Daddi,and Margrethe Wold.Emanuele Daddi is especially thanked for providing the ERO count mod-els used in this paper.

References

Bergstr¨o m,S.,Wiklind,T.2004,A&A,414,95

Bertin,E.&Arnouts,S.1996,A&AS,117,393 Bolzonella,M.,Miralles,J.-M.&Pell′o,R.2000,A&A,363, 476

Bruzual,G.&Charlot,S.1993,ApJ,405,538

Chabrier,G.,Bara?e,I.,Allard,F.,Hauschildt,P.2000,ApJ, 542,464

Cimatti,A.,Villani,D.,Pozzetti,L,di Serego Alighieri,S.

2000,MNRAS,318,453

Cimatti,A.,Daddi,E.,Mignoli,M.et al.2002,A&A,381,L68 Cimatti,A.,Daddi,E.,Cassata,P.et al.2003,A&A,412,L1 Cohen,M.1994,AJ,107,582

Cole,A.,Lacey,C.G.,Baugh,C.M.,Frenk,C.S.2000,MNRAS, 319,204

Cruz,K.L.,Reid,I.N.,Liebert,J.;Kirkpatrick,J.D.,Lowrance, P.J.2003,AJ,126,2421

Daddi,E.,Cimatti,A.,Pozzetti,L.et al.2000,A&A,361,535 Daddi,E.,Cimatti,A.,Renzini,L.2000,A&A,362,L45 Daddi,E,.Cimatti,A.,Broadhurst,T.,et al.2002,A&A,384, L1

Eggen,O.J.,Lynden-Bell,D.&Sandage,A.R.1962,ApJ,136, 748

Elbaz,D.&Cesarsky,C.2003,Science,300,270

Firth, A.E.,Somerville,R.S.,McMahon,R.G.et al.2002 MNRAS,332,617

Fukugita,M.,Shimasaku,K.&Ichikawa,T.1995,PASP,107, 945

Fukugita,M.,Ichikawa,T.,Gunn,J.E.et al.1996,AJ,111, 1748

Gilbank, D.G.,Smail,I.,Ivison,R.J.,Packham, C.,2003, MNRAS,submitted(astro-ph/0308318)

Gonzalez-Solares,E.A.,Perez-Fournon,I.,Rowan-Robinson, M.et al.2004,MNRAS,submitted(astro-ph/0402406) Hall,P.B.&Green,R.F.1998,ApJ,507,558

Irwin,M.&Lewis,J.2001,NewAR,45,105

Kau?mann,G.&Charlot,S.1998,MNRAS,297,L23 Kochanek,C.S.,Pahre,M.A.,Falco,E.E.et al.2001,ApJ,560, 566

Landolt,A.U.1992,AJ,104,340

Larson,R.B.1975,MNRAS,173,671

Mann,R.G,Oliver,S.,Carballo,R.,et al.2002,MNRAS,332, 549

Martini,P.2001,AJ,121,2301

Marzke,R.O.,Geller,M.J.,Huchra,J.P.,Corwin,H.G1994, AJ,108,437

McMahon,R.G.,Walton,N.A.,Irwin,M.J.et al.2001, NewAR,45,97

Mohan,N.R,Cimatti,A.,R¨o ttgering,H.J.2002,A&A,383, 440

Moriondo,G.,Cimatti,A.,Daddi,E.2000,A&A,364,26 Moustakas,L.A.,Casertano,S.,Conselice,C.,et al.,2004,ApJ, 600,L131

Oliver,S.,Rowan-Robinson,M.et al.2000,MNRAS,316,749 Pozzetti,L.,Mannucci,F.2000,MNRAS,317,L17

Pierini, D.,Maraston, C.,Bender,R.,Witt, A.N.2004, MNRAS,347,1

Roche,N.D.,Almaini,O.,Dunlop,J.S.,Ivison,R.J.&Willott,

C.J.2002,MNRAS,337,1282

12V¨a is¨a nen&Johansson:Bright EROs and massive galaxies at z～1 Roche,N.D.,Dunlop,J.S.,Almaini,O.2003,MNRAS,346,

803

Rowan-Robinson,M.,Lari,C.,Perez-Fournon,I.,et al.2004,

MNRAS,in press(astro-ph/0308283)

Saracco,P.,Longhetti,M.,Severgnini,P.,et al.2003,A&A,

398,127

Scodeggio,M.&Silva,D.R.2000,A&A,359,953

Scott,S.E.,Fox,M.J.,Dunlop,J.S.et al.2002,MNRAS,331,

817

Silva,L.,Granato,G.L.,Bressan,A.,Danese,L.1998,ApJ,

509,103

Silva,L.,1999,PhD thesis,SISSA(International School for

Advanced Studies),Trieste,Italy.

Smail,I.,Owen,F.N.,Morrison,G.E.,et al.2002,ApJ,581,

844

Smith,G.P.,Smail,I.,Kneib,J.-P.,et al.2002,MNRAS,330,

Somerville,R.S.,Primack,J.R.1999,MNRAS,310,1087

Takata,T.,Kashikawa,N.,Nakanishi,K.,et al.2003,PASJ,

55,789

V¨a is¨a nen,P.,Tollestrup,E.V.,Willner,S.P.,Cohen,M.2000,

ApJ,540,593(V00)

V¨a is¨a nen,P.,Johansson,P.2004,A&A,submitted

White,S.D.M.&Rees,M.J.1978,MNRAS,183,341

White,S.D.M.&Frenk,C.S.1991,ApJ,379,52

Wold,M.,Armus,L.,Neugebauer,G.,Jarrett,T.H.,Lehnert,

M.D.2003,AJ,126,1776

Yan,L.&Thompson,D.2003,ApJ,586,765

Yan,L.,Thompson,D.,Soifer,B.T.2003,AJ,127,1274

V¨a is¨a nen&Johansson:Bright EROs and massive galaxies at z～113 RA(J2000)DEC(J2000)g′r′i′Z J K

1816h06m52.7s54?46′37.3′′>25.0022.49±0.0521.06±0.0320.29±0.0318.75±0.1916.89±0.20

1916h08m15.6s54?41′51.7′′>25.0022.60±0.0620.94±0.0320.02±0.0319.03±0.2317.07±0.17

2016h08m38.7s55?00′19.5′′>25.0023.14±0.0921.52±0.0420.45±0.0618.88±0.2417.29±0.26

2116h08m59.4s54?56′34.1′′24.72±0.1722.84±0.0721.33±0.0320.08±0.0419.06±0.2017.10±0.18

2216h08m60.0s54?56′25.7′′>25.0022.84±0.0721.41±0.0420.17±0.0519.16±0.2317.12±0.19

2316h10m21.0s55?07′28.1′′24.72±0.1522.86±0.0821.14±0.0320.32±0.0518.83±0.1717.24±0.24

2416h10m30.3s55?08′21.1′′24.58±0.1322.73±0.0721.47±0.0320.99±0.0918.83±0.1917.18±0.20

2516h10m40.4s55?05′25.7′′24.31±0.1122.72±0.0721.16±0.0320.20±0.0418.88±0.2317.18±0.21

2616h10m51.0s55?12′05.8′′24.29±0.1122.99±0.0821.75±0.0421.04±0.0919.31±0.3317.47±0.24

2716h10m59.1s54?31′26.2′′>25.0022.90±0.0721.28±0.0420.15±0.0319.00±0.5517.25±0.23

2816h11m08.9s54?50′17.7′′>25.0023.07±0.0721.64±0.0520.85±0.0619.42±0.3117.42±0.24

2916h11m12.8s55?08′24.7′′23.43±0.0522.57±0.0621.25±0.0320.17±0.0418.73±0.2416.91±0.17

3016h11m14.7s55?05′31.8′′>25.0023.02±0.0921.43±0.0320.75±0.0719.09±0.2617.07±0.21

3116h11m58.4s54?53′56.5′′>25.0022.91±0.0821.50±0.0420.50±0.0518.93±0.2417.40±0.23

3216h34m14.4s41?14′42.5′′24.49±0.1622.32±0.0420.71±0.0319.91±0.0318.81±0.2716.59±0.26

3316h34m22.8s40?56′07.1′′>25.0022.76±0.0620.96±0.0420.23±0.0419.13±0.2617.12±0.23

3416h34m52.1s40?50′50.2′′>25.0023.02±0.0821.41±0.0520.57±0.0519.05±0.2317.39±0.20

3516h34m58.4s40?52′55.4′′>25.0022.80±0.0621.10±0.0420.36±0.0418.86±0.1917.20±0.16

3616h35m12.1s40?47′31.0′′>25.0023.31±0.1121.82±0.0820.69±0.0719.13±0.2717.60±0.25

3716h36m01.0s41?05′59.3′′>25.0022.99±0.0821.58±0.0620.30±0.0519.01±0.1717.22±0.21

3816h36m07.5s41?21′42.3′′24.69±0.0022.39±0.0420.97±0.0019.95±0.0019.11±0.2716.74±0.23

3916h36m07.8s41?03′41.4′′>25.0022.94±0.0721.35±0.0720.78±0.0719.56±0.2717.24±0.23

4016h36m58.1s40?49′43.9′′>25.0022.41±0.0520.74±0.0319.95±0.0318.64±0.2016.72±0.11

4116h37m23.9s41?01′16.0′′>25.0023.27±0.1021.42±0.0920.66±0.0519.19±0.2717.64±0.26

4216h37m27.5s40?55′43.6′′>25.0023.03±0.0021.27±0.0020.90±0.0818.97±0.2417.45±0.26

4316h37m31.0s40?53′36.3′′23.93±0.1022.47±0.0520.76±0.0319.99±0.0318.57±0.2016.79±0.16

4416h37m38.6s40?56′59.3′′>25.0023.00±0.0721.36±0.0520.39±0.0419.19±0.4017.10±0.17

4516h37m40.2s40?54′21.8′′>25.0022.64±0.0520.80±0.0519.66±0.0518.86±0.2317.10±0.20

4616h37m49.6s40?55′43.5′′>25.0022.66±0.0520.86±0.0320.12±0.0318.92±0.2416.98±0.17

4716h37m57.2s41?19′38.4′′24.18±0.0023.09±0.0821.57±0.0620.75±0.0819.65±0.3117.45±0.22

4816h37m58.1s41?09′47.6′′24.24±0.1222.80±0.0621.48±0.0520.56±0.0619.40±0.2817.15±0.19

4916h38m20.3s41?03′42.5′′23.73±0.0822.21±0.0420.55±0.0219.76±0.0318.72±0.3116.62±0.23

5016h38m46.1s41?08′18.8′′>25.0023.37±0.1021.64±0.0620.48±0.0619.39±0.2917.27±0.24

Table1.Photometry of the whole ERO sample.Sources1–17are those with r?K>5.5and i?K>4.4,while 18–50and51–54are selected by only r?K>5.5and i?K<4.4,respectively.

14V¨a is¨a nen&Johansson:Bright EROs and massive galaxies at z～1

Sample N N?Area arcmin2K limEROs ERO/arcmin2 (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) Table2.Columns(2)to(5)give the total numbers of sources and stellar objects,the surveyed area,and the average limiting magnitude.The rest of the columns refer to galaxies only:columns(6)to(7)give mean colours,though note that these are lower limits since optical non-detections we ignored.Column(8)gives the number of EROs selected by r?K≥5.5,(9)by i?K≥4.4,and(10)by their combination;column(11)is the range of surface densities resulting from either r or i band based selection;column(12)is the range of cumulative surface densities at K<17

r?K>5.5r?K>5.8i?K>4.4

K limit Frac.ΣK Frac.ΣK Frac.ΣK

(mag)N%arcmin?2N%arcmin?2N%arcmin?2

Table3.The sample of Mt.Hopkins EROs.Cumulative counts are given for di?erent ERO colour selection criteria. Total numbers(N),Fraction of EROs compared to the total galaxy count(“Frac.”)is calculated using galaxy counts in these same?elds(V¨a is¨a nen et al.2000).At K≤17.5N shows the completeness corrected value,and the raw count is given in parentheses.

对数函数及其性质重点难点创新突破

对数函数及其性质重点难点创新一、教学目标课程标准对本节课的要求为：理解对数函数的概念及单调性，掌握对数函数的图象通过特殊点，依据学生的学习基础及自身特点结合课标要求，我确定了本节课的教学目标：知识目标：1、理解对数函数的定义，掌握对数函数的图象和性质； 2、会求和对数函数有关的函数的定义域； 3、会利用对数函数单调性比较两个对数的大小。能力目标：1、通过对底数的讨论，使学生对分类讨论的思想有进一步的认识，体会由特殊到一般的数学思想； 2、通过例题、习题的解决，使学生领悟化归思想在解决问题中的作用。情感目标：学生在参与中感受数学，探索数学，提高学习数学的兴趣，增强学好数学的自信心。二、教学重难点：教学重点：理解对数函数的定义，掌握对数函数图象和性质；教学难点：底数a对函数值变化的影响及对数函数性质的应用。三`教学方法：通过让学生观察、思考、交流、讨论、发现对数函数的图象的特点四、课堂结构设计：本节课是概念、图象及性质的新授课，为了使学生更好的达成学习目标我设计了以学生活动为主体，以培养学生能力为中心，提高课堂教学质量为目标的课堂结构。这是我的课堂结构设计：

五、教学媒体设计：根据本节课的教学任务和学生学习的需要，我设计了利用多媒体课件展示引例、例题、习题和练习……，增大教学的容量，也使学生易于接受，提高学生的学习兴趣和积极性；利用几何画板演示作图，展示图象的动态变化过程，有效地突出重点、突破难点、提高教学效率，增强直观性和准确性。这是我的教学媒体设计：钟 15 分钟钟钟 6 分钟

六、教学过程设计在对教材及学生全面深入了解的基础上，我设计了以下五个教学环节：

函数图象重难点分析

函数图象重难点分析本资料为woRD文档，请点击下载地址下载全文下载地址www.5y kj.co m 用“五点法”画函数的简图，及函数，，的图像与正弦曲线的联系，参变数A，对图像的影响是本课的重点．弄清函数与图像的关系，特别是和对图形的影响是本课学生的一个难点．克服难点的办法，是要让学生弄清：（1）在函数中，对函数性质所起的作用；（2）函数的图像是通过怎样的方法由正弦曲线变化而得到，三个参数在图像变换中起什么作用．本节运用了对图像的三种变换：振幅变换，是由A的变化引起的；周期变换，是由的变化引起的；相位变换（也叫沿x轴方向的平移变换）：是由的变化引起的．将函数图像与各点的横坐标不变，纵坐扩大到原来的2倍，得到的图像，将图像上各点的纵标不变，横坐标扩大到原来的2倍，得到的图像．在这里，学生往往弄不明白为什么沿y轴“扩大到2倍”是乘以2，沿x轴“扩大到2倍”

却是除以2？函数图像在横纵两个坐标轴上的拉伸为什么不一致．也弄不明白在横纵两轴的平移究竟是什么样子．其实这些问题在学生们学习了坐标轴的变换及曲线与方程的关系后很容易理解．我们可以通过“点变换”去认识“线变换”．（1）的图像与的图像上横坐标相同的相应两点与之间的关系要满足，可见，将图像点横坐标不变，纵坐标伸长（A>1）或压缩（A<1）到原来的A倍，变成了的图像．（2）的图像与的图像纵坐标相同的相应两点和，之间的关系要满足，即，因此，将图像上各点的纵坐标不变，横坐标压缩或伸长到原来的倍，就变成了的图像．可以用类似的方法解释为什么时，把的图像向左平移个单位得到的图像，而时，要把的图像向右平移个单位得到的图像．在图像变换的教学中，要教给学生利用观察、对比、分析找出变换的规律，弄清变换的原因，理解变换的过程，而不能死记变换的结论．特别要掌握“变换”中的辩证观点：由点变换认识线变换． “五点法”作图，是作函数的静态图，在学习初期对了解函数图像的形状有益，继续学习时，必须从国家的变换角度研究图像间的关系，也就是要教给学生在运动变化中，寻

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《对数函数》教学设计一、教材分析本小节选自《中等职业教育课程改革国家规划新教材-数学（基础模块上册）》第四章，主要内容是学习对数函数的定义、图象、性质及初步应用。对数函数是继指数函数之后的又一个重要初等函数，无论从知识或思想方法的角度对数函数与指数函数都有许多类似之处。与指数函数相比，对数函数所涉及的知识更丰富、方法更灵活，能力要求也更高。学习对数函数是对指数函数知识和方法的巩固、深化和提高，也为解决函数综合问题及其在实际上的应用奠定良好的基础。二、学生学习情况分析刚从初中升入高一的学生，仍保留着初中生许多学习特点，能力发展正处于形象思维向抽象思维转折阶段，但更注重形象思维。由于函数概念十分抽象，又以对数运算为基础，同时，初中函数教学要求降低，初中生运算能力有所下降，这双重问题增加了对数函数教学的难度。教师必须认识到这一点，教学中要控制要求的拔高，关注学习过程。三、设计理念本节课以建构主义基本理论为指导，以新课标基本理念为依据进行设计的，针对学生的学习背景，对数函数的教学首先要挖掘其知识背景贴近学生实际，其次，激发学生的学习热情，把学习的主动权交给学生，为他们提供自主探究、合作交流的机会，确实改变学生的学习方式。四、教学目标 1．通过具体实例，直观了解对数函数模型所刻画的数量关系，初步理解对数函数的概念，体会对数函数是一类重要的函数模型； 2．能借助计算器或计算机画出具体对数函数的图象，探索并了解对数函数的单调性与特殊点； 3．通过比较、对照的方法，引导学生结合图象类比指数函数，探索研究对数函数的性质，培养学生运用函数的观点解决实际问题。五、教学重点与难点重点是掌握对数函数的图象和性质，难点是底数对对数函数值变化的影响．六、教学过程设计教学流程：背景材料→引出课题→函数图象→函数性质→问题解决→归纳小结（一）熟悉背景、引入课题 1．让学生看材料：如图1材料（多媒体）：某种细胞分裂时，由1个分裂成2个，2个分裂成4个……，