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Pixelated Lenses and H 0from Time-delay QSOs Liliya L.R.Williams Department of Physics and Astronomy University of Victoria Victoria,BC,V8P 1A1,Canada and Prasenjit Saha 1Department of Physics Oxford University Keble Road,Oxford,OX13RH,UK Received
ABSTRACT
Observed time delays between images of a lensed QSO lead to the determination of the Hubble constant by Refsdal’s method,provided the mass distribution in the lensing galaxy is reasonably well known.Since the two or four QSO images usually observed are woefully inadequate by themselves to provide a unique reconstruction of the galaxy mass,most previous reconstructions have been limited to simple parameterized models,which may lead to large systematic errors in the derived H0by failing to consider enough possibilities for the mass distribution of the lens.We use non-parametric modeling of galaxy lenses to better explore physically plausible but not overly constrained galaxy mass maps,all of which reproduce the lensing observables exactly,and derive the corresponding distribution of H0’s.Blind tests—where one of us simulated galaxy lenses,lensing observables,and a value for H0,and the other applied our modeling technique to estimate H0indicate that our procedure is reliable.For four simulated lensed QSOs the distribution of inferred H0have an uncertainty of?10%at90%con?dence.Application to published observations of the two best constrained time-delay lenses,PG1115+080and B1608+656,yields H0=61±11km sec?1Mpc?1at68%con?dence and61±18km sec?1Mpc?1at 90%con?dence.
1.Introduction
Most ways of measuring the Hubble constant involve a form of distance ladder,which utilizes a number of astrophysical standard candle and standard ruler relations,and is calibrated locally by a geometrical technique such as parallax(e.g.,Madore et al.1999, Madore et al.1998,Kennicutt1995).A recent exciting development in this?eld is to extend the reach of the geometrical rung of the distance ladder by using masers in orbit around galaxy centers to get distances to nearby galaxies thus bypassing Cepheids(Herrnstein
et al.1999).A few methods involve no distance ladder:good examples are(i)inferring the distance of Type II supernovae from their light curves and spectra by modeling their expanding photospheres(Schmidt et al.1992),and(ii)comparing the H0-independent angular extent of galaxy clusters to their H0-dependent depth as deduced by the X-ray emission,and the Sunyaev-Zeldovich microwave background decrement due to the cluster (Hughes&Birkinshaw1998).
But the most‘one-step’method of all was proposed by S.Refsdal in1964,though it has only recently become feasible.The principle of Refsdal’s method is simple.In a system where a high-redshift QSO is split into multiple images by an intervening galaxy lens,the di?erence in light travel time between di?erent images(observable as time delays if the QSO is variable)is proportional to the scale factor of the universe.The time delay is given by the schematic formula
Time delay =h?1× 1month × image separation in arcsec 2
×z lens× weak dependence on z lens,z QSO,and cosmology (1)
× lens-mass-distribution dependent factor
where the last two factors are of order unity.To obtain H0using this method one requires three types of input:(i)the observed time delay(s)between QSO images,(ii)knowledge of the cosmology,and(iii)the mass distribution in the lensing galaxy.The?rst can and has been measured with increasing precision for about eight systems so far.The second is not a serious problem,because the dependence on cosmology is weak and the uncertainty due to it is easy to quantify;in this paper we will refer all results to the Einstein-de Sitter cosmology.The uncertainty in H0is dominated by the third item;the number of usable constraints on the mass distribution in the galaxy are few,while the range of possible distributions is huge.Thus,mass distribution is the major source of uncertainty.
Two di?erent paths can be taken to compensate for our lack of knowledge about the galaxy.One is to assume an exact parametric form for the galaxy mass distribution and?t the observed lensing properties as best as possible;the other is to take the image properties as exact,and try to reconstruct the galaxy mass map as best as possible.Single parametric models?x the last term in(2)and thus cannot account for the uncertainty resulting from it. Blandford&Kundi′c(1996)advise that even if one?nds a parametric galaxy model which is dynamically possible and which reproduces the image properties with acceptably lowχ2,
one still has to‘aggressively explore all other classes of models’to get the true uncertainty in H0.To explore the model space in a systematic fashion one needs to use a representation of the galaxy mass distribution that is general and not restricted to a particular form.One way would be to expand the mass distribution using a set of basis functions,another is to pixelate the galaxy and take each pixel as an independent mass element.We introduced pixelated models in Saha&Williams(1997,hereafter SW97)but at that time did not have any strategy for searching model space.We have now extended that work to explore the model space with the goal of estimating the uncertainty in the derived value of H0.
The plan of this paper is as follows.In Section2we summarize the observational situation with regard to strongly lensed QSOs.In Section3we present the general lensing formalism and point out a few properties of the lensing equations that are useful in interpreting the results of modeling.We also explain the reasons for con?ning our analysis to PG1115+080and B1608+656for now.Sections4and5describe our method for deriving H0and test it on a synthetic sample via a blind test.Application to the real systems can be found in Section6.Section7discusses our results.
2.Observed Time-Delay Lenses
The?rst piece of input for H0determination is the measurement of time delays between the various QSO images.At the present time,ten multiply-imaged QSOs already have measured time delays or are being monitored:Q0957+561(Kundi′c et al.1997a), PG1115+080(Schechter et al.1997,Barkana1997),B1608+656(Fassnacht et al.1999), B0218+257(Biggs et al.1999),PKS1830-211(Lovell et al.1998),HE1104-1805(Wisotzki et al.1998),B1030+074,B1600+434(Burud et al.1999),J1933+503,and RXJ0911+0551 (Hjorth et al.1999).In this work we limit ourselves to4-image lenses with known source and lens redshifts and accurate time delay measurements;PG1115+080and B1608+656?t the description.
PG1115(Weymann et al.1980)was the second lens to be discovered.The source is a radio-quiet QSO at z s=1.722.Accurate positions for the images were measured by Kristian et al.(1993);lightcurves were analyzed by Schechter et al.(1997),and time delays derived by Schechter et al.and Barkana(1997).The main lensing galaxy is an outlying member of a small galaxy group,z l=0.311with an estimated line of sight velocity dispersion of270±70km s?1(Kundi′c et al.1997b).A summary of observational results on this system can be found in SW97.
B1608was discovered in the Cosmic Lens All-Sky Survey(Myers et al.1995,Myers et al.1999).The lens is either a perturbed single galaxy or a merging/interacting pair of galaxies superimposed in the plane of the sky.The source and lens redshifts are1.394and 0.630respectively.The time delays were recently reported by Fassnacht et al.(1999)based on VLA observations spanning7months.The time delays we use in this work are an earlier
determination(Fassnacht,private communication),and are less than0.5σaway from the values quoted in Fassnacht et al.(1999);?t BA=28.5,?t BC=32,and?t BD=77.
3.Lensing formalism
A photon traveling through a galaxy will take longer to arrive at the observer then an unimpeded photon.Part of the time delay occurs because the path of the ray bundle makes a detour rather than going straight;the time delay is further increased because the photon travels through the gravitational potential well of the galaxy.The total time delay is given
by,
τ(θ,θs)=(1+z l)D ol D os
2
(θ?θs)2?
1
?θ=0=θ?θs?
1
|θ?θ′|2
(3)
Image distortion and magni?cation are given by the inverse of the curvature matrix of the arrival time surface ?2τ
location.In such a situation the total arrival time con?guration is centrally symmetric and the resulting images are approximately equidistant from the galaxy center.
(3)If the four images of a single source are located at di?erent galactocentric distances the simplest explanation is the presence of external shear.External shear e?ectively raises the gravitational part of the arrival time surface closest to itself(see Fig.1and Eq.2).The e?ect is to push the locations of the stationary points away from the source of external shear,hence increasing the radial spread of images.It follows that the direction of external shear can be determined by examining image locations with respect to the galaxy center. PG1115is a good example;the image closest to the galaxy center,image B,is located between the galaxy-lens and the galaxy group,which is the source of external shear in this case.
(4)Position angles(PA)of images are determined by the ellipticity PA of the galaxy roughly at the radius of the images.When images are spread over a range of radial distances their PA provide information on galaxy ellipticity PA over a range of galactocentric distances.Thus detailed modeling can reveal the twisting of the isodensity contours.
(5)Not all types of information about images are equally useful for modeling purposes. The arrival time surface integrates overκ(θ)twice,making time delays most sensitive to the overall mass distribution in the galaxy,and least dependent on the local small-scale perturbations in the mass distribution.Image positions are determined from the lensing equation which integrates overκ(θ)once.Finally,image magni?cations are very dependent on the local behavior of mass,making them the least useful for modeling.This means, unfortunately,that a double like Q0957,though it has well-measured substructure in the images and near-perfect time-delay measurements,provides too few constraints on the lensing mass to usefully estimate H0unless drastic assumptions about the mass distribution are made.In that case,the derived errors will tend to be underestimated as was noted by Bernstein and Fischer(1999)who constructed many types of parametric models for Q0957:‘The bounds on H0are strongly dependent on our assumptions about a“reasonable”galaxy pro?le’.
(6)A linear rescaling of the arrival time and lens equations,i.e.,multiplying both by a constant factor?will not alter the observable properties of images,image separations and relative magni?cation tensors.Physically the transformation amounts to rescaling the mass density of the lens by?and adding a constant mass density sheet.This transformation was?rst discussed by Gorenstein et al.(1988)with regard to modeling of Q0957,and later became known as the mass sheet degeneracy.Note that a mass sheet extending to in?nity is not needed,a mass disk larger than the observed?eld is enough because an external monopole has no observable e?ect.
4.The method
The?rst step is to pixelate the lens plane mass distribution of the main lensing galaxy. In practice we use～0.1′′pixels,and limit the galaxy to a circular window of radius about twice that of the image-ring.Pixelated versions of Eqs.(2)and(3)are:
τ(θ,θs)=(1+z l)D ol D os
2
|θ|2?θ·θs? nκnψn(θ) (5)
and
θ?θs? nκn αn(θ)=0,(6) where the summation is over mass pixels andψn andαn are integrals over individual pixels and can be evaluated analytically(see Appendix of SW97).A term|θs|2has been omitted from Eq.(3)because a constant additive factor in the arrival time cannot be measured.
Image properties translate into linear constraints in the(N+2)-dimensional model space,where N dimensions represent a pixel each and2represent source coordinates.We call these primary constraints.The images can provide us with only a few constraints:in a4-image system we have2×4coordinates and3time delay ratios:11in all.On the other hand,the unknowns are numerous,～202mass pixels plus2source coordinates.This results in a plethora of galaxy models each of which reproduces the image properties exactly. Luckily,the bulk of these models can be discarded because they do not look anything like galaxies.In fact,we consider only those models which satisfy the following further(linear) constraints,which we call secondary.These pertain to the main lensing galaxy:
1.mass pixel values,κn must be non-negative;
2.the location of the galaxy center is assumed to be coincident with that of the
optical/IR image;
3.the density gradient of the lens must point no more than45?away from the center of
the galaxy;
4.the lens must have inversion symmetry,i.e.,look the same if rotated by180?[enforced
only if the main lensing galaxy appears unperturbed and has no companions close to the QSO images];
5.logarithmic projected density gradient in the vicinity of the images,
d lnκ/d lnθ=ind(r)should b
e no shallower that?0.5.For a power law
projected density pro?le,radial magni?cation at an image is equal to?1/ind(r), therefore a statement that ind(r)
6.external shear,i.e.,in?uence of mass other than the main lensing galaxy is restricted
to be constant across the image region,and is represented by adding a term
1
5.Blind tests of the method
Before applying the method to real systems we try it on a synthetic situation designed to resemble the real world as close as possible.One of us,“Person A”,picked an h value and created a set of four galaxies and the corresponding images of a single background source in each case.Exact values of image positions with respect to the galaxy center and time delays(but not h,nor information as to whether the galaxy was inversion symmetric or if there was any external shear)were conveyed to the other one of us,“Person B”,who used this information to construct an ensemble of galaxy models and derive h distributions for each case separately.
We ran the whole experiment several times to remove bugs,and did not want to fall into the trap of simply publishing the results of the best run.So once we were con?dent that the experiment worked,we decided that the next four galaxies,whatever the results, would go into the published paper.Figure2pictorially illustrates the three stages of our blind test.
Person B applied the reconstruction method to each system twice,once with the assumption of inversion symmetry(i.e.,symmetric galaxies,see item4in Section4),and once without.Based on the appearance of the reconstructed mass distribution Person B decided whether inversion symmetry constraint was right in each case.Figures3–10present the results for each of the four galaxies.For galaxies#1,3and4Person B picked symmetric options,and the asymmetric option for galaxy#2.Panels(a)and(b)of Figures3,5,7,and 9show the actual projected density distribution and the average of the100reconstructed galaxies,for galaxies#1,2,3,and4respectively.In a map which is an average of many reconstructions,persistent features of individual maps are enhanced while peculiarities are washed out,so the average is a reasonable guess as to what the real galaxy looks like,in a probabilistic sense.
Panels(a)of Figures4,6,8,and10plot the slope of density pro?le,ind(r)vs.derived h.The‘real’value of h is0.025.In all the cases the slope of the density pro?le,ind(r) in the vicinity of the images correlates with the derived h value,though the degree of correlation and its slope is not universal.Qualitatively,the reason for the correlation is easily understood.A relatively?at galaxy density pro?le,i.e.,|ind(r)|is small,translates into a?at gravitational contribution to the arrival time surface,and‘?lls’the well of the geometrical time delay contribution evenly resulting in small?uctuations in the amplitude of the total arrival time surface.Thus the predicted time delays between images will be small,and to keep the observed time delays?xed the derived h has to be small as well.
Panels(b)of Figures4,6,8,and10show the derived h probability distribution. These distributions look di?erent for all galaxies,because galaxy morphologies are di?erent. Since all four are independent probability distributions based on that galaxy,the overall distribution is just the product of the four,see Figure11.The solid histogram is the product of the four distributions presented in panels(b)of Figures4,6,8,and10.The
dashed histogram is similar,but results from Person B excluding what appeared to be the
best constrained galaxy(#3),and the dotted histogram represents the case where inversion
symmetry was not applied to any of the systems.All three resultant distributions recover h fairly well,with the90%of the models contained within20%of the true h.However
the distributions are not the same;the most probable values are di?erent by～10%.This illustrates how a relatively minor feature in modeling constraints,namely exclusion or
inclusion of inversion symmetry,can make a considerable di?erence in the estimated h value
when the goal is to achieve precision of10%.Based on this observation we conclude that assumed galaxy shape in parametric reconstructions plays a major role in determining the
outcome of the H0determination.
How robust are the results to the changes in other modeling assumptions?Changing
pixel size by a factor of～1.5,and relaxing mass gradient angle constraint(item3in Section4)does not change our results considerably.
6.Application to real systems
6.1.PG1115
Figure12shows the results of the reconstruction for PG1115.Since the main lensing
galaxy has no close companions and its light pro?le is smooth we have included inversion symmetry as one of the modeling constraints.The average of100arrival time surfaces is
shown in Figure12(a);Figure12(b)shows the corresponding caustics and critical lines. The latter are not as smooth as the former because locations of caustics and critical lines
are derived using the gradients the arrival time surface,which are always noisier than the
original function.Panels(c)and(d)plot the quantityθ·θs? nκnψn(θ),and the total arrival time surface,respectively.The plot of the modi?ed gravitational potential,(c),
illustrates the e?ect of external shear which is due to a galaxy group to the lower right of
the main galaxy.
Because ind(r)has been measured for the main lensing galaxy in PG1115,the relation between pro?le slope and derived H0,Figure13(a),can be used to derive an upper limit on H0.Impey et al.(1998)?t the galaxy light with a de Vaucouleurs pro?le of an e?ective radius r e=0.59′′.At the location of the images,about1.3′′from galaxy center the double logarithmic density slope is ind(r)=?2.3.Assuming that the mass pro?le can only be shallower than the light pro?le,and consulting Figure13(a)we place an upper limit on H0 of75km sec?1Mpc?1.
If the true mass density pro?le slope is isothermal the corresponding H0is
30km sec?1Mpc?1.A low value of H0was also obtained by parametric models that assumed
isothermal models for the galaxy(Schechter et al1997).
In the blind test,Section5,we assumed that all time delays are known precisely,which is not currently the case for any of the systems except Q0957.What e?ect does an error in time delay determination have on the derived H0?Figure13(b)shows two distributions derived using two di?erent?t determinations based on the same lightcurves.There is a 20%di?erence in the most probable value of H0in the two histograms,but overall they are not very di?erent.Both distributions are very broad;90%of the models span the range between30and75km sec?1Mpc?1.
Figure14shows a dense version of the arrival time surface.The regions of the plot where the contours are sparse are the?attest,i.e.,most‘stationary’regions of the lens plane.This is where one would expect to?nd images of sources placed close to the main source.For example if the point-like QSO is surrounded by a host galaxy the image of that galaxy will be well delineated by these‘empty’regions.In fact,the observed optical ring in the case of PG1115is well reproduced by the ring in Figure14.
6.2.B1608
The light distribution of the lensing system is rather messy,possibly representing
a merging/interacting galaxy pair,therefore inversion symmetry was not used in the following reconstructions.Figures15and16are the same as Figures12and13,but for B1608.The range50to100km sec?1Mpc?1in Figure16(b)encompasses about90%of the reconstructions.
https://www.doczj.com/doc/f18915530.html,bined p(h)plot
Just like in the case of the blind test we now multiply probability distributions from PG1115and B1608to get the combined distribution,Figure17.90%of all points lie within the range43–79km sec?1Mpc?1,while the median of the distribution is61km sec?1Mpc?1. Note that the errorbars obtained using our method are substantially larger than what is usually quoted in other studies.We ascribe this increase to the more systematic sampling of the whole image-de?ned model space unrestricted by the con?nes of parametric models.
7.Discussion and Conclusions
Multiply-imaged QSO systems provide us with an elegant way of measuring H0,and a lot of observational and modeling e?ort has been invested in this enterprise.As the quality of the observational data improves,most of the uncertainty in H0is contributed by the mass distribution in the lens.How to treat this problem is a matter of some debate.Should
one use a single,physically motivated mass model,or should one approach the problem with no preconceptions about the galaxy form?
In general,ad hoc restrictions on the allowed mass models translate into a too optimistic and probably biased estimated distribution of H0’s.To avoid this trap one has to allow as much freedom for the lens models as possible.On the other hand,to yield a useful estimate of H0one has to restrict the amount of freedom allowed for the models using physically motivated criteria.Ideally one wants to balance these two opposing tendencies and impose just the correct quantity and quality of model constraints.Based on our experience from the present work we conclude that parametric,or any other approach that severely restricts the freedom of the galaxy-lens,has over-constrained their models and thus ended up with unrealistically small errorbars,and biased H0’s.As a result di?erent models of the same systems can yield discrepant results.For example,Romanowski&Kochanek (1998)use dynamical methods to model the galaxy in Q0957,and further constrain the
galaxy to be similar to nearby ellipticals;they quote61+13
?15at2σlevel.Bernstein&Fischer
(1999)analyzed the same system but used a range of astrophysically reasonable parametric
models.Their estimate,77+29
?24,also at2σlevel,does not agree with that of Romanowski&
Kochanek.Our approach is di?erent in that it does not presuppose a galaxy shape,but instead allows us to impose as many or as few constraints as is deemed appropriate.
The most unrestricted models would be constrained solely by what we call the primary constraints,i.e.,image observables.By de?nition these would yield unbiased estimates of H0based on lensing data alone.We chose to go somewhat beyond this and apply what we call secondary constraints,which describe realistic galaxies in most general terms.The derived H0distributions are narrower;the price we pay is a small amount of bias.It can be argued that we are still too generous with our mass models,i.e.other galaxy characteristics can be safely assumed,and hence tighter constraints can be applied to the models without sacri?cing the unbiased nature of results.This avenue can be taken in the future work if additional constraints become available.
A potential source of additional modeling constraints are optical rings,lensed images of the QSO galaxy host,which are seen in some cases,for example,in PG1115and B0218. The orientations and elongations of individual images of QSO host galaxy can be used as linear inequality constraints to narrow down the range of possible galaxy mass distributions.
If two or more sources with known redshifts were lensed by the same foreground galaxy, these could be used to break the mass sheet degeneracy and thus further constrain the galaxy.However in practice cases of two sources at di?erent redshifts lensed by the same galaxy are expected to be very rare because of the small galaxy cross-sections.
Probably the most promising potential constraint is based on the relation between the slope of the projected density pro?le around the images and the derived H0.If the slope can be estimated by means other than lensing,or at least a limit placed on its value as we did in Section6.1using the observed slope of the light distribution,then H0can be
constrained much better compared to what is currently possible.
With the two systems used in the present work,PG1115+080and B1608+656,and implementing primary constraints of image properties and secondary constraints describing a few general properties of lensing galaxies,we conclude that H0is between43and
79km sec?1Mpc?1at90%con?dence level,with the best estimate being61km sec?1Mpc?1.
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Fig. 1.—Generic arrival-time surface for a four-image QSO(in fact,a possible one for PG1115).The black?lled circles mark the images and the gray?lled circle marks the source, while the curves are isochronal contours.This is only one of six possible con?gurations for a4+1image system(Section5.5of Schneider et al.1992),but the most natural one where the lens is very roughly circular.The top image is the lowest minimum in the arrival-time surface and hence the?rst image.Next comes the minimum at lower left,followed closely by the saddle point at left,and?nally the saddle point at lower right.The contours indicate one more image—a maximum close to the source—but this is not seen in real systems;it is presumed to be demagni?ed below visibility because of the centrally peaked mass distribution in lensing galaxies.
Fig.2.—Pictorial illustration of the three stages of the blind tests designed to quantify the trustworthiness of our H0estimation method.
Fig.3.—(a)The actual mass pro?le of galaxy#1.There is external shear in this case;its source is located to the upper right of the galaxy,as can be deduced from the con?guration of the images with respect to the center of the main lensing galaxy.(b)An average of100
reconstructed mass pro?les.In both the panels the contours are at1
3,1,...,in units of
critical density for lensing.The four solid dots are the locations of the images.
Fig.4.—(a)Slope of the projected density pro?le,d lnκ/d lnθ=ind(r)plotted against the derived h value.(b)Probability distribution of derived h values from galaxy#1.The actual value of h is0.025for all four blind-test galaxies.The dot indicates the location of the‘most isothermal’reconstructed galaxy.Note that the whole estimated range spans a factor of3 in h,and the distribution is not Gaussian.
Fig.5.—Same as Figure3but for galaxy#2.The source of external shear is located to the upper right of the main galaxy.
Fig.6.—Same as Figure4,but for galaxy#2.
Fig.7.—Same as Figure3but for galaxy#3.There is no external shear in this case.
Fig.8.—Same as Figure4,but for galaxy#3.
WHEN与WHILE用法区别 when, while这三个词都有"当……时候"之意，但用法有所不同，使用时要特别注意。 ①when意为"在……时刻或时期"，它可兼指"时间点"与"时间段"，所引导的从句的动词既可以是终止性动词，也可是持续性动词。如： When I got home, he was having supper.我到家时，他正在吃饭。 When I was young, I liked dancing.我年轻时喜欢跳舞。 ②while只指"时间段"，不指"时间点"，从句的动词只限于持续性动词。如：While I slept, a thief broke in.在我睡觉时,盗贼闯了进来。 辨析 ①when从句与主句动作先后发生时，不能与while互换。如： When he has finished his work, he takes a short rest.每当他做完工作后，总要稍稍休息一下。(when = after) When I got to the cinema, the film had already begun.当我到电影院时，电影已经开始了。(when=before) ②when从句动词为终止性动词时，不能由while替换。如： When he came yesterday, we were playing basketball.昨天他来时，我们正在打篮球。 ③当从句的谓语是表动作的延续性动词时，when, while才有可能互相替代。如：While / When we were still laughing, the teacher came in.正当我们仍在大声嬉笑时，老师进来了。 ④当从句的谓语动词是终止性动词，而且主句的谓语动词也是终止性动词 时,when可和as通用，而且用as比用when在时间上更为紧凑，有"正当这时"的含义。如： He came just as (or when) I reached the door.我刚到门那儿，他就来了。 ⑤从句的谓语动词如表示状态时，通常用while。如： We must strike while the iron is hot.我们应该趁热打铁。 ⑥while和when都可以用作并列连词。
when 和 where 引导定语从句的用法 定语从句既是英语语法的一个重点，同时又是一个难点。说它是难点，主要难在两点上：一点是如何正确判断什么样的汉语句子要译为英语的带定语从句的复合句；另一点是定语从句的引导词较多(包括关系代词who, that, which, as 和关系副词when, where, why)，而且其用法也较复杂。那么到底什么情况下用when和where来引导定语从句呢？它们又该怎么用呢？下面就举例说明： 一、when：当主句中的先行词(即主句中被后面定语从句修饰的词)是表示时间意义的名词时，它只能作定语从句的时间状语，放在定语从句句首。如果定语从句的引导词是作该定语从句的主语或宾语，则要改用关系代词that或which来引导。例如： The days when we used foreign oil are gone. 我们用洋油的日子一去不复返了。 I'll never forget the day when I was born. (=I'll never forget my birthday.) 我永远不会忘记我出生的日子。 It happened in November when the weather was wet and cold. 这事发生在天气又湿又冷的十一月。 In the years that (which) followed, Marx kept on studying English and using it. 在这之后的几年中，马克思继续学习和使用英语。(that作定语从句"that followed"的主语) The day (that) I always remember in all my life is my birthday. 我一生中最难忘的日子是我的生日。(that作定语从句"that I always remember in all my life"的宾语，that可以省略)
一、从句是如何出题的？ 1. 时态 2. 考连接词 3. 考语言顺序 二、学好从句的两个基本条件 1. 时态 2. 从句的三个必须:①必须是句子；②必须有连接词；③必须是陈述句 三、状语从句、宾语从句、定语从句重点 1.如何判断何种从句 2. 从句的时态 3. 从句的连接词与扩展 4. 经典单选、从句与选词、长句子分析 四、如何判断三种从句 1. 状语从句无先行词 2. 宾语(表语)从句无先行词有动词或词组 3. 定语从句先行词多为名词或代词 一、When引导的定语从句与时间状语从句的区分 1. when的译法不同。在时间状语中，when 翻译成“当……的时候” I want to be a teacher when I grow up. 当我长大的时候，我要做一名老师。在定语从句中，when不翻译。I won't forget the day when he says he loves me. 我不会忘记他说爱我的那一天。 2. 在时间状语中，when从句前面或后面是句子；定语从句中，when 从句不能位于句首，且通常when前为表示时间的名词day、year等。 3. when在从句的作用不同。在时间状语从句中，when是连词，只起连接主句和从句的作用，不做从句的任何成分。不过when引导的时间状语从句修饰主句的谓语，做主句的时间状语。 在定语从句中，when是关系副词，在从句中代替先行词做从句的时间状语，修饰从句的谓语。 例1 I will always remember the days when I lived with my
grandparents in the country. 例2 I always remember the days in the country when I see the photo of my grandparents. 点评：例1意为“我会永远记得跟我祖父母一起住在乡下的那些日子”，其中when 引导的是一个定语从句, 修饰the days, when在从句中作时间状语。例2意为“当我看到祖父母的照片时，总是会想起在乡下的那些日子”，其中when 引导的从句并不修饰前面的名词the country，因此可判定为时间状语从句。 例1中的when可用in which替代，即从句可改为...in which I lived with my grandparents in the country. 例2中从句前有名词，但根据句意可 知并不是从句所修饰的对象，也不能用“介词+ which”来替代。 二、判断关系代词与关系副词 方法一：用关系代词，还是关系副词完全取决于从句中的谓语动词。及物动词后面无宾语，就必须要求用关系代词；而不及物动词则要求用关系副词。例如： This is the mountain village where I stayed last year. 这是我去年呆过的山村。 I'll never forget the days when I worked together with you. 我永远不会忘记与你共事的日子。 判断改错： 1. This is the mountain village where I visited last year. 2. I will never forget the days when I spent in the countryside. 3. This is the mountain village （which）I visited last year.
when和while用法区别专项练习 讲解三例句： 1. The girls are dancing while the boys are singing. 2. Kangkang’s mother is cooking when he gets home. 3. When/While Kangkang’s mother is cooking, he gets home. 一、用when或者while填空 Margo was talking on the phone, her sister walked in. we visited the school, the children were playing games. · Sarah was at the barber’s, I was going to class. I saw Carlos, he was wearing a green shirt. Allen was cleaning his room, the phone rang. Rita bought her new dog; it was wearing a little coat. 7. He was driving along ________ suddenly a woman appeared. 8. _____ Jake was waiting at the door, an old woman called to him. 9. He was reading a book ______suddenly the telephone rang. 10. ______ it began to rain, they were playing chess. [ 二、用所给动词适当形式填空 11. While Jake __________ (look) for customers, he _______ (see) a woman. 12. They __________ (play) football on the playground when it _____ (begin) to rain. 13. A strange box ________ (arrive) while we _________ (talk). 14. John ____________ (sleep) when someone __________ (steal) his car. 15. Father still (sleep) when I (get) up yesterday morning. 16. Grandma (cook) breakfast while I (wash) my face this morning. 17. Mother (sweep) the floor when I (leave) home. ~ 18. I (read) a history book when someone (knock ) at the door. 19. Mary and Alice are busy (do) their homework. 20. The teacher asked us (keep) the windows closed. 21. I followed it (see) where it was going. 22. The students (play) basketball on the playground from 3 to 4 yesterday afternoon. / 三、完成下面句子，词数不限 1.飞机在伦敦起飞时正在下雨。 It when the plane in London. 2.你记得汶川大地震时你在做什么吗 Do you remember what you when Wenchuan Earthquake . 3.当铃声想起的时候，我们正在操场上玩得很开心。 We on the playground when the bell . 4.当妈妈下班回家时，你在做什么 % when Mum from work 5.当我在做作业时，有人敲门。 I was doing my homework, someone
过去进行时过去进行时表示过去某一时刻或者某段时间正在进行或发生的动作，常和表过去的时间状语连用，如： 1. I was doing my homework at this time yesterday. 昨天的这个时候我正在做作业。 2. They were waiting for you yesterday. 他们昨天一直在等你。 3. He was cooking in the kitchen at 12 o'clock yesterday. 昨天12点，他正在厨房烧饭。 过去进行时的构成： 肯定形式：主语+was/were+V-ing 否定形式：主语+was not (wasn't)/were not (weren't)+V-ing 疑问形式：Was/Were+主语+V-ing。 基本用法： 1. 过去进行时表示过去某一段时间或某一时刻正在进行的动作。常与之连用的时间状语有，at that time/moment, (at) this time yesterday (last night/Sunday/week…), at+点钟+yesterday (last night / Sunday…)，when sb. did sth.等时间状语从句，如： 1）What were you doing at 7p.m. yesterday? 昨天晚上七点你在干什么？ 2）I first met Mary three years ago. She was working at a radio shop at the time. 我第一次遇到玛丽是在三年前，当时她在一家无线电商店工作。 3）I was cooking when she knocked at the door. 她敲门时我正在做饭。 2. when后通常用表示暂短性动词，while后通常用表示持续性动词，而while所引导的状语从句中，谓语动词常用进行时态，如： When the car exploded I was walking past it. = While I was walking past the car it exploded. 3. when用作并列连词时，主句常用进行时态，从句则用一般过去时，表示主句动作发生的过程中，另一个意想不到的动作发生了。如： I was walking in the street when someone called me. 我正在街上走时突然有人喊我。 4. when作并列连词，表示“（这时）突然”之意时，第一个并列分句用过去进行时，when引导的并列分句用一般过去时。如：
Period____8______ in Unit ___6_________ （Period_____20_____Week 4-6）Subject _Grammar (B) Teaching aims To use the Past Continuous Tense with while and when correctly Teaching interesting points of this class: The differences of when and while in Past Continuous Tense Teaching steps: I. Cooperation and intercourse 1. Check the preparation 2. Discuss in groups II. Question and explanation Check the exercises on page 101, and explain the differences between when and while. when和while的区别是： when只能用于一般时态 while可以用于进行时态 when conj. 在...的时候, 当…的时候 when 在绝大多数情况下，所引导的从句中，应该使用非延续性动词（也叫瞬间动词） 例如：I'll call you when I get there. 我一到那里就给你打电话 I was about to go out when the telephone rang.我刚要出门，电话铃就响了。 但是，when 却可以be 连用。 例如：I lived in this village when I was a boy. 当我还是个孩子的时候我住在这个村庄里。 When I was young, I was sick all the time. 在我小时候我总是生病 while 当...的时候 While he was eating, I asked him to lend me $2. 当他正在吃饭时,我请他借给我二美元。While I read, she sang. 我看书时，她在唱歌。 while 的这种用法一般都和延续性动词连用 while 可以表示“对比‘，这样用有的语法书认为是并列连词 Some people like coffee, while others like tea. 有些人喜欢咖啡, 而有些人喜欢茶。 as 当...之时，一边......一边....... I slipped on the ice as I ran home. 我跑回家时在冰上滑了一跤 He dropped the glass as he stood up. 他站起来时，把杯子摔了。 She sang As she worked. 她一边工作一边唱歌。 III. Rumination and evaluation when, as, while这三个词都可以引出时间状语从句，它们的差别是：when 从句表示某时刻或一段时间as 从句表示进展过程，while 只表示一段时间 When he left the house, I was sitting in the garden. 当他离开家时，我正在院子里坐着。 When he arrived home, it was just nine o'clock.
when 和while的用法区别 ①when是at or during the time that, 既指时间点，也可指一段时间； while是during the time that，只指一段时间，因此when引导的时间状语从句中的动词可以是终止性动词，也可以是延续性动词，而while从句中的动词必须是延续性动词。 ②when 说明从句的动作和主句的动作可以是同时，也可以是先后发生；while 则强调主句的动作在从句动作的发生的过程中或主从句两个动作同时发生。 ③由when引导的时间状语从句，主句用过去进行时，从句应用一般过去时；如果从句和主句的动作同时发生，两句都用过去进行时的时候，多用while引导，如： a. When the teacher came in, we were talking. 当此句改变主从句的位置时，则为： While we were talking, the teacher came in. b. They were singing while we were dancing. ④when和while 还可作并列连词。when表示“在那时”；while表示“而，却”，表对照关系。如： a. The children were running to move the bag of rice when they heard the sound of a motor bike. 孩子们正要跑过去搬开那袋米，这时他们听到了摩托车的声音。 b. He is strong while his brother is weak. 他长得很结实，而他弟弟却很瘦弱。 when，while，as引导时间状语从句的区别 when，while，as显然都可以引导时间状语从句，但用法区别非常大。 一、when可以和延续性动词连用，也可以和短暂性动词连用；而while和as只能和延续性动词连用。 ①Why do you want a new job when youve got such a good one already？（get 为短暂性动词）你已经找到如此好的工作，为何还想再找新的？ ②Sorry，I was out when you called me．（call为短暂性动词）对不起，你打电话时我刚好外出了。 ③Strike while the iron is hot．（is为延续性动词，表示一种持续的状态）趁热打铁。 ④The students took notes as they listened．（listen为延续性动词）学生们边听课边做笔记。 二、when从句的谓语动词可以在主句谓语动作之前、之后或同时发生；while和as从句的谓语动作必须是和主句谓语动作同时发生。 1．从句动作在主句动作前发生，只用when。 ①When he had finished his homework，he took a short rest．（finished先发生）当他完成作业后，他休息了一会儿。 ②When I got to the airport，the guests had left．（got to后发生）当我赶到飞机场时，客人们已经离开了。 2．从句动作和主句动作同时发生，且从句动作为延续性动词时，when，while，as都可使用。
While和When在过去进行时中两者的区别如下： ①when是at or during the time that, 既指时间点，也可指一段时间；while是during the time that，只指一段时间，因此when引导的时间状语从句中的动词可以是终止性动词，也可以是延续性动词，而while 从句中的动词必须是延续性动词。 ②when 说明从句的动作和主句的动作可以是同时，也可以是先后发生；while 则强调主句的动作在从句动作的发生的过程中或主从句两个动作同时发生。 ③由when引导的时间状语从句，主句用过去进行时，从句应用一般过去时；如果从句和主句的动作同时发生，两句都用过去进行时的时候，多用while引导，如： a. When the teacher came in, we were talking. 当此句改变主从句的位置时，则为： While we were talking, the teacher came in. b. They were singing while we were dancing. ④when和while 还可作并列连词。when表示“在那时”；while表示“而，却”，表对照关系。如： a. The children were running to move the bag of rice when they heard the sound of a motor bike. 孩子们正要跑过去搬开那袋米，这时他们听到了摩托车的声音。 b. He is strong while his brother is weak. 他长得很结实，而他弟弟却很瘦弱。 when; while 当……时候 while能用when代替; 但是when却不一定能用while代替. while+从句, 动作一定会延续 when+延续性动词/瞬间动词; when he arrived
1、When 和While 的区别 when,while 都有“当……时候”的意思. when 既可表示某一点时间,也可以表示某一段时间. （一）在when 引导的时间状语从句中,其谓语动词可以是延续性的,也可以是非延续性的,可与主句中的谓语动词同时发生,也可在其后发生. 例如： 1、I was just reading a book when she came into my room. 她走进我房间时,我正在看书. （二）、while 只能表示某一段时间,不能表示某一点时间.在while 引导的时间状语从句中,其谓语动词只能是延续性的,而且也只能与主句中的谓语动词同时发生或存在. 例如： 1、While Jim was mending his bike, Lin Tao came to see him. 正当吉姆修自行车时,林涛来看他. Remark: 由when 引导的时间状语从句，主句用过去进行时，从句应用一般过去时；如果从句和主句的动作同时发生，两句都用过去进行时的时候，多用while 引导，如： a. When the teacher came in, we were talking. 2、倒装知识点 完全倒装 表示地点的副词here, there 置于句首, 且主语是名词(不是代词) 时 表示时间、方向的副词或介词短语置于句首, 且主语是名词(不是代词) 时 作表语的形容词、分词、介词短语、such 置于句首时 部分倒装 “ only+状语” 置于句首, 主句需要部分倒装 具有否定意义或半否定意义的副词以及含否定词的介词短语置于句首作状语时“so 或neither + 助动词/情态动词/be 动词+主语”表示“……也/也不” so/such...that...句型 以had/were/should 开头省略if 的虚拟条件句
过去进行时 过去进行时表示过去某一时刻或者某段时间正在进行或发生的动作，常和表过去的时间状语连用，如： 1. I was doing my homework at this time yesterday. 昨天的这个时候我正在做作业。 2. They were waiting for you yesterday. 他们昨天一直在等你。 3. He was cooking in the kitchen at 12 o'clock yesterday. 昨天12点，他正在厨房烧饭。 过去进行时的构成： 肯定形式：主语+was/were+V-ing 否定形式：主语+was not (wasn't)/were not (weren't)+V-ing 疑问形式：Was/Were+主语+V-ing。 基本用法： 1. 过去进行时表示过去某一段时间或某一时刻正在进行的动作。常与之连用的时间状语有，at that time/moment, (at) this time yesterday (last night/Sunday/week…), at+点钟+yesterday (last night / Sunday…)，when sb. did sth.等时间状语从句，如： 1）What were you doing at 7p.m. yesterday? 昨天晚上七点你在干什么？ 2）I first met Mary three years ago. She was working at a radio shop at the time. 我第一次遇到玛丽是在三年前，当时她在一家无线电商店工作。 3）I was cooking when she knocked at the door. 她敲门时我正在做饭。 2. when后通常用表示暂短性动词，while后通常用表示持续性动词，而while所引导的状语从句中，谓语动词常用进行时态，如： When the car exploded I was walking past it. = While I was walking past the car it exploded. 3. when用作并列连词时，主句常用进行时态，从句则用一般过去时，表示主句动作发生的过程中，另一个意想不到的动作发生了。如： I was walking in the street when someone called me. 我正在街上走时突然有人喊我。 4. when作并列连词，表示“（这时）突然”之意时，第一个并列分句用过去进行时，when引导的并列分句用一般过去时。如： 1）I was taking a walk when I met him. 我正在散步，突然遇见了他。 2）We were playing outside when it began to rain. 我们正在外边玩，这时下起雨来了。 过去进行时和一般过去时的用法比较： 1）过去进行时表示过去正在进行的动作，而一般过去时则表示一个完整的动作。 例如：They were writing letters to their friends last night. 昨晚他们在写信给他们的朋友。（没有说明信是否写完） They wrote letters to their friends last night.
as when while 的区别和用法 as when while的用法 一、as的意思是“正当……时候”,它既可表示一个具体的时间点，也可以表示一段时间。as可表示主句和从句的动作同时发生或同时持续，即“点点重合”“线线重合”；又可表示一个动作发生在另一个动作的持续过程中，即“点线重合”， 但不能表示两个动作一前一后发生。如果主句和从句的谓语动词都表示持续性的动作，二者均可用进行时，也可以一个用进行时，一个用一般时或者都用一般时。 1、As I got on the bus，he got off. 我上车，他下车。（点点重合）两个动作都是非延续性的 2、He was writing as I was reading. 我看书时，他在写字。（线线重合）两个动作都是延续性的 3、The students were talking as the teacher came in. 老师进来时，学生们正在讲话。（点线重合）前一个动作是延续性的，而后一个动作时非延续性的 二、while的意思是“在……同时（at the same time that )”“在……期间(for as long as, during the time that）”。从while的本身词义来看，它只能表示一段时间，不能表示具体的时间点。在时间上可以是“线线重合”或“点线重合”，但不能表示“点点重合”。例如： 1、He was watching TV while she was cooking. 她做饭时，他在看电视。（线线重合） 2、He was waiting for me while I was working. 我工作的时候，他正等着我。（线线重合） 3、He asked me a question while I was speaking. 我在讲话时，他问了我一个问题。（点线重合）
when和while的用法和区别 while和when都是表示同时，到底句子中是用when还是while主要看从句和主句中所使用的动词是短暂性动作（瞬时动词）还是持续性动作。 1、若主句表示的是一个短暂性的动作，而从句表示的是一个持续性动作时，两者都可用。如：He fell asleep when [while] he was reading. 他看书时睡着了。 I met him when [while] I was taking a walk in the park. 我在公园散步时遇到了他。 2、若主、从句表示两个同时进行的持续性动作，且强调主句表示的动作延续到从句所指的整个时间，通常要用while。如： Don’t talk while you’re eating. 吃饭时不要说话。 I kept silent while he was writing. 在他写的时候，我默不作声。 3、若从句是一个短暂性动作，而主句是一个持续性动作，可以when 但不用while。如：When he came in, I was listening to the radio. 他进来时，我在听收音机。 It was raining hard when we arrived. 我们到达时正下着大雨。 4、若主、从句表示的是两个同时(或几乎同时)发生的短暂性动作，一般要用when。如： I thought of it just when you opened your mouth. 就在你要说的时候，我也想到了。 至于什么是短暂性动作，什么是持续性动作，其实有个很简单的规律。就是如果是进行时态，一般是持续性的。如果是过去式，一般是短暂性动作。 对于，填写when还是while的问题，通常首先看主句和从句中的时态，再根据以上4个规律来判断填写那个单词。 When和While的区别 通常when 后面接一般过去时while 后面接过去进行时 ①when是at or during the time that, 既指时间点，也可指一段时间，while是during the time that，只指一段时间，因此when引导的时间状语从句中的动词可以是终止性动词，也可以是延续性动词，而while从句中的动词必须是延续性动词。 ②when 说明从句的动作和主句的动作可以是同时，也可以是先后发生；while 则强调主句的动作在从句动作的发生的过程中或主从句两个动作同时发生。 ③由when引导的时间状语从句，主句用过去进行时，从句应用一般过去时；如果从句和主句的动作同时发生，两句都用过去进行时的时候，多用while引导，如： a. When the teacher came in, we were talking. 当此句改变主从句的位置时，则为： While we were talking, the teacher came in. b. They were singing while we were dancing. ④when和while 还可作并列连词。when表示“在那时”；while表示“而，却”，表对照关系。如： a. The children were running to move the bag of rice when they heard the sound of a motor bike.
when 的用法 一、when 用作副词。 1. 用作疑问副词，引导特殊疑问句。(什么时候，何时 [at what time ])例如： ①When will you come to see me? ②When are they going to visit the Great Wall? 2. 用作连接副词，通常用来引导名词性从句[主语从句、表语从句、同位语从句、宾语从句]及起名词作用的“when ＋动词不定式”结构。(什么时候，何时 [at which; on which ])例如： ①When he comes is not known. [主语从句] ②The morning is when I am busiest. [表语从句] ④I don't know when the plane takes off. [宾语从句] ⑤I don't know when to leave for London. [宾语] 3. 用作关系副词，引导限定性定语从句和非限定性定语从句。(在…的时候 [at/on/in/during which])例如： ①Do you still remember the days when we stayed in America? ②The day will come soon when the Chinese astronauts will go to the moon. ③It happened ten years ago, when I was a child. ④We will go to the countryside at the beginning of June, when the summer harvest will start. 二 . when 用作连词。 1. 用作从属连词，意为“当……的时候[at the time when ]”，引导时间状语从句。例如： ①They learned a lot from the peasants when they stayed in the village. ②It was snowing when he arrived at the station. 【点津】如果 when 引导的从句中的主语和主句中的主语一致，且从句中的谓语动词是“be ＋分词”或从句主语是 it ，则 be 动词及其主语常可省略。例如： ③When（ he was ）asked why he was late, he made no answer. ④I'll tell him about it when（ it is ）possible. 2. 用作从属连词，意为“一……就……”，引导时间状语从句。[immediately after]例如： ①We will stand up when the teacher comes into the classroom. 老师一进教室我们就起立。 ②Fire the rockets when I give the signal. 3. 用作从属连词，意为“还没 / 刚刚/刚一……就”，引导时间状语从句。[immediately after]例如： ①I had hardly opened the door when he came in. 我刚一开门，他就进来了。 ②I had not been reading for half an hour when I heard someone call my name. 【点津】hardly…when和no sooner…than的结构要注意三点：意思为“一A就B“；A句通常用完成时态；hardly 和 no sooner 位于句首时要注意部分倒装。 4. 用作从属连词，意为“倘若，如果”，表示条件。例如： ①Turn off the switch when anything goes wrong with the machine. 如果机器发生故障，就把电源关上。 ②He will be likely to recover when he is operated on. 5. 用作从属连词，意为“既然，尽管”，表示让步。例如： ①Why use metal when you can use plastic?既然能用塑料,为什么用金属? ②They kept trying when they knew it was hopeless. 尽管他们知道那件事没有希望，可是他们还在不断地努力。6. 用作并列连词，意为“在那时，届时;就在这时”，表示时间。这时主句中可以用过去进行时，过去完成时或“ was/were about to do sth. ”结构。 ①Last night I was about to go to bed when the phone rang. ②I was cooking in the kitchen when someone knocked at the door. ③He had just finished the book when supper was served. 7. 用作并列连词，意为“虽然、然而、可是”，表示转折。例如： ①He usually walks to work when he might take a bus. 虽然他可 以坐公共汽车上班，但他却常常步行上班。 ②I had only twenty dollars when I needed thirty to buy the dictionary.我需要 30 美元买那本字典，可是我只有20美元。 8. 用作并列连词，意为“而、却”，表示对比。例如： How can he say that everything is fine when it's obvious that it is not? 他怎能说一切都好呢？情况显然不是那样。 三 . when 用作代词[which time]。 when 作为代词常常位于介词之后，意为“那时，什么时候”。例如： ①Since when have you been studying Japanese? 【点津】since when 作引导词时是“介词（since）+关系代词（when）”的结构，when意为which time。since when常引导非限制性定语从句，从句应用完成时态。 ②I came here in 1949, since when I have been engaged in this work. 我1949年到这里，从那时起我就担任这项工作。 ③We came back on Tuesday, since when we have been working in the repair shop. ④We came a week ago, since when the weather has been bad. 四、when 还可用作名词，前面常常用定冠词 the。 the when 表示事 件发生的时间,常常与 the where, the how 并列使用。 He told the police the when and the how of the accident. 他告 诉警察事故发生的时间及发生的原委。 ［巩固练习］ 1. I remember ______ this used to be a quiet village. A. when B. how C. where D. what 2. Why do you want a new job _____ you've got such a good one already? A. that B. where C. which D. when 3. It was an exciting moment for these football fans this year, ______ for the first time in years their team won the World Cup. A. that B. while C. which D. when 4. We are living in an age _______ many things are done on computer. A. which B. that C. where D. when 5. The reporter said that the UFO ______ east to west when he saw it. A. was travelling B. travelled C. had been travelling D. was to travel 6. ______ got into the room _________ the telephone rang. A. He hardly had; then B. Hardly had he; when C. He had not; than D. Not had he; when 7. ---- Can I join your club, Dad? ---- You can when you _______ a bit older. A. get B. will got C. are getting D. will have got 8. I shall never forget those days ________ I lived in the countryside with the farmers, ________ had a great effect on my life. A. that; which B. when; which C. which; that D. when; who 9. The film brought the hours back to me ______ I was taken good care of in that faraway village. A. until B. that C. when D. where 10. He was about to tell me the secret ______ someone patted him on the shoulder.