Mini Microscopes

Last year, I became curious about the options for cheap, but good microscopes. I expected to find some decent equipment available for approximately $100, with more powerful equipment available at higher price ranges.

To my surprise, the options are better and cheaper than I expected.

The $2.50 Microscope. Cell phone not included.

First, I discovered that moderate-power microscopes that attach to a cell phone (and make use of its built-in camera) are remarkably cheap. At the time of writing, a quick search of Amazon indicates that the cost of a 60x microscope that attaches to a cell phone is about $2.50. These little gizmos are extremely portable, project via the cell phone display to allow multiple people to view at the same time, and are just good enough to resolve large plant cells. This was an amazing find for the money, and allows a lot of fun, educational experiences for little effort.

To my even greater surprise, I later found out that significantly more powerful magnification can be produced for even cheaper.

The Foldscope project, run by Manu Prakash at Stanford University, strives to make powerful (and durable) microscopes available at ultra-low prices for clinical and educational use around the world. The premise is this: A microscope with incredible resolution (sufficient for viewing bacteria, and approaching the absolute limit that is possible for an optical microscope) can be assembled in minutes from parts costing under $1! Moreover, the resulting device is ridiculously portable and resistant to damage, able to survive abuse that would pulverize a typical microscope. (And, if you do manage to destroy it, so what? Build another one from scratch for under $1.)

How is this possible? Obviously, the answer is simplicity. While a conventional microscope has a pair of lenses built into an optical path with precision engineering, a Foldscope has only one lens and the user controls the focus by manually adjusting the position of the Foldscope and its parts from the eye.

The 40¢ Microscope

There is one other catch: The recipe for making a Foldscope is not completely fleshed out. Currently, there is a beta test in progress. Instructions for assembling a Foldscope have been published, but these instructions are not quite ready for prime time. And so, last spring, I resolved to wait until Prakash’s group had produced a “final” set of Foldscope instructions.

But I was only willing to wait so long. In October, I downloaded some slightly-contradictory sets of instructions, ordered the parts online, and sat down to make myself a Foldscope. At first, I spent over an hour carefully trimming parts out of a printout (carefully-folded paper is the main structural component of a Foldscope, hence the name). Then I needed to mount a lens, which was not very well described in any of the materials before me. So I improvised, sandwiching a 2.4mm borosilicate sphere (the lens) between two small pieces of printer paper, held tight using Elmer’s Glue.

Paramecium Photographed with the 40¢ Microscope

And lo, it worked. Without even using the rest of the Foldscope that I had so carefully sliced out a printout, this tiny, postage-stamp-sized microscope worked. Here’s the tale of the tape: 140x magnification for 40¢. All I had to do was press it against a prepared slide (stained onion epidermis was my first test item), hold it close to my eye, look up at a light and fiddle around with the distances between my eye, the lens, and the sample. Within seconds, I found a good focus and could clearly see the cell nuclei. A few minutes later, I was clearly seeing human blood cells, and by holding the whole setup against my cell phone camera, I was taking pictures of these samples. Amazing!

One remarkable thing about mini microscopes is this: Hardly anybody realizes that they exist, or even that they can exist. I've shown them to medical doctors, a microbiologist, the coordinator of science at a private school, and many other well-educated adults working in technical fields; they were all astonished that such a thing is possible. Maybe one day, people will carry a microscope everywhere they go. This could happen today. For various reasons, the use cases and usability have not come together to start a revolution in widespread microscope use. Maybe that will change someday soon.

HL Tau

A remarkable image was released by the NRAO today. This image, made using data collected with the ALMA radio astronomy facility in Chile, shows the protoplanetary disk of dust around the star HL Tau (clickable at right), as imaged in radio wavelengths with resolution that is unprecedented for such a formation in space.

What does it show? The bright disk, as seen from a moderately oblique angle is dust and smaller particles, warm because of illumination from the host star. The dark lanes in the disk are the fascinating thing, though, because they indicate that larger bodies are using their gravitational influence to sweep up the dust at certain radii from the star. Those bodies are planets, and while they are too small to be seen in the image, their handiwork is evident.

What is wonderful about this image is how inclusive it is. Extrasolar planets have been discovered around many stars, but there is no particular indication, when one or more planets are discovered orbiting another star, which other planets might exist without having been observed. This image, however, shows the whole disk, and any large planet would have to clear a lane, so we are seeing the whole system to a pretty good resolution, and this includes at least eight clear lanes, plus a ninth if you count the outside of the last dusty region.

Does this mean there are eight or nine planets? Perhaps. There are at least two ways a planet could clear a lane. One, by being in the lane. In that case, the planet would sweep up dust that comes close to it, and scatter away some dust, to leave its immediate vicinity relative dust-free. A second way that a planet could clear a lane is to clear a lane that is some distance away from it, where the ratio of orbital periods between the planet and dust particles is the ratio of small integers (2:1, 3:2, etc.), and then the planet could clear a distant lane by tugging its dust particles away, a little at a time.

To investigate this possibility, I quickly checked the period ratios that are implied by the distances of the dark lanes from the star. Counting out from the center to the edge, with gap A closest to the star and I representing the outer boundary of the disk, these periods are (in arbitrary units):

A  0.12, B 0.44, C 0.67, D 0.91, E 1.29, F 1.67, G 2.27, H 2.73, I 3.34

By and large, the ratios of these periods are pretty sparing in compelling small-integer ratios. The ratio C:B is about 3:2, D:C is about 4:3, and I:F is about 2:1, but otherwise, the ratios seem pretty arbitrary and therefore seemingly unrelated to long-distance lane clearing. This could mean that all or almost all of the clear lanes represent single bodies, which is an exciting possibility, because it would mean a system of 8 (or almost 8) planets in orbits that are currently non-overlapping and therefore might continue to represent individual planets as the system evolves and all the dust is swept up by the planets.

If these are eight planets, they are distributed quite differently from our solar system, which has much larger distances between the outer planets than the inner ones. Perhaps what we see at HL Tau are the cores of what will eventually be larger planets, with the accretion of smaller, inner planets yet to become evident.

Whatever the details yet to be discerned, this remarkable observation indicates that we are gaining the ability to map other solar systems, at least those with easily-seen dust disks, in remarkable detail.

Kepler 186-f: Another Earth?

Today, Kepler 186-f is being announced as the first planet discovered orbiting another star that shares the same basic properties as Earth. This is a monumental discovery in science that fulfills one of the Kepler mission’s major goals and brings us further down the path towards even bigger discoveries. This is the Kepler candidate I listed seventh one week ago on a list of possible earthlike planets, and now it’s the first one to be announced as confirmed.

What prompted this news?

Kepler “sees” planets as the star they orbit dims slightly when planets pass in front of them, blocking a tiny fraction of its light. These signals are hard to pick out, so discoveries begin as candidates, when we think they might be real, and are called confirmed when further investigation indicates they are (almost) certainly real.

We already knew that this discovery, if it turned out to be real, looked pretty earthlike, which is why it was high on my list. The news indicates that it is indeed a real planet.

What do we know about it?

We know that its star, a red dwarf (class M), is smaller and cooler than the Sun (class G). We know that the planet orbits its star every 130 days and is one of at least five planets orbiting the star. It orbits the star at about 40% the Earth’s distance from the Sun, which is the same distance that Mercury is from our Sun. But because its star is much cooler than the Sun, the amount of heat that the planet receives is about the same as the Earth does.

The estimated size of the planet is just a bit bigger than the Earth. For now, however, there is considerable uncertainty in that estimate, so it may turn out to be significantly bigger or smaller.

We have no way at present of estimating what Kepler 186-f’s climate might be like. It could be much hotter or colder than Earth, lack the kind of atmosphere that Earth has, or be suffocating under a much thicker atmosphere. We won’t know until we gather more data about planets of this size and temperature how often they evolve to be more like Venus (too hot), Mars (too cold), Earth (just right), or something else.

Can a red dwarf star support an earthlike planet?

There is speculation that a red dwarf might be a bad place for an earth-sized planet to become earthlike in other ways, because the planet has to orbit close-in to get enough warmth. At that distance, the tides that the star causes on the planet might cause an excess of volcanic activity, and/or force the planet’s rotation to keep one side always facing the star in eternal day and the other side in eternal night. Either of those things could make the development of life difficult or even impossible. But, that is all speculative, and in any case, Kepler 186-f orbits out at a relatively long period of 130 days, which is longer than Mercury’s. Mercury rotates in synch with its orbit in an interesting way, but it doesn’t have that eternal day-night divide, nor does it have tidally-powered volcanoes, so Kepler 186-f may escape that fate.

When will we know more?

We probably know about as much about Kepler 186-f now as we knew about Mars four hundred years ago, before the invention of the telescope. If we could see Kepler 186-f through a telescope, we could learn a lot about it, but that’s a huge challenge because it’s 500 light years away, and so close to its star that our best telescopes couldn’t even separate it from the star, which is millions of times brighter than the planet.

That distance also means that even if a spacecraft left tomorrow to go visit it, going at 99% the speed of light, it would still take 1000 years for us to get the data from that mission, so that’s simply not an option.

A future telescope, superior to any that have ever yet been planned, might be able to give us more information, perhaps by studying the light from its entire system and subtracting the light we receive when Kepler 186-f is behind its star from the light we receive when it is not.

The good news, however, is that we expect many other stars to have planets like this, and many of those stars will be much closer to us than Kepler 186-f. By the time we have a telescope that could perform more detailed scientific studies of Kepler 186-f, we will almost certainly know about many more planets to look at, and return more detailed information about the ones that are closer than 500 light years.

This discovery, however, should excite interest in the construction of such telescopes, which have been proposed but not approved.

Could Kepler 186-f have intelligent life?

The prevalence of extraterrestrial intelligence is something we can only speculate about today, but efforts like the SETI project, which search for radio signals that an extraterrestrial civilization might be sending, probably have no better single target than Kepler 186-f. All we can do is point our radio telescopes that way, listen, and wait.

Has Kepler Found Other Earths?

Earthlike Planets: Found or Not Found?

For years now, headlines in major newspapers have announced that the Kepler spacecraft has found earthlike planets. These headlines are misleading, trading on the vagueness of the terms “earthlike”, “find”, and “discover”. In fact, Kepler has found no single case known to be very much like the Earth, but rather several cases which may turn out, pending further study, to be earthlike. The purpose of this post is to survey the most promising cases, and what they collectively say about the prevalence of earthlike planets.

Kepler data can be used to determine the size of a planet and how much light it receives from its star. Different researchers have used different standards of how close to Earth’s parameters one chooses to define as earthlike. I will follow some of the more stringent definitions in calling earthlike a size of 0.8–1.25 R⊕ (Earth radius) and an equilibrium temperature (the temperature one would expect given the amount of light it receives from its star) of 185K–303K. It must be noted, though, that Kepler’s measurements of size and temperature have considerable uncertainty. The error (one standard deviation) in size is typically close to half the estimated size, and the error in temperature is often greater than the difference between Death Valley and Antarctica. Therefore, I will include for consideration planets with estimated size 0.7–1.6 R⊕ and estimated equilibrium temperatures from 160K–325K because better measurements may reveal some of these planets to fall in the tighter ranges.

Another kind of uncertainty is whether or not the planet actually exists! Kepler observes the dimming of a star which may be due to a planet transiting in front of it, but false positives (FPs) may be due to other explanations:

Electronic False Positives: Noise in the Kepler instrument may cause random dimming of a star which coincidentally resembles the transits of an earthlike planet. I discussed in my last post how this occurs and how we might diagnose such cases.

Astrophysical False Positives: A real astrophysical eclipse or transit may be taking place involving a background star which coincidentally is aligned so as to be observed by the same pixel in Kepler’s instrument as the brighter star. A real Kepler discovery is shown schematically in case (a) in this figure, whereas cases (b) and (c) show two examples of astrophysical false positives that can look the same as (a) in Kepler data.

The Sample

Kepler discoveries, initiated with algorithmic searches of the data, and proceeding to human inspection of the most promising candidates, are called Kepler Objects of Interest (KOIs). After examination, a KOI may be classified as Confirmed (nearly certain to exist), Candidate (may exist), False Positive (does not exist), or Not Dispositioned (uncertain status at this time). For a sample of sunlike stars, we take the 144,308 Kepler targets with a magnitude of 16.0 or brighter and logg (a measure of density) of 4.0–4.9.

Completeness

It is important to understand the completeness of Kepler's search for planets: For any given star in the sample that does happen to have an earthlike planet circling it, the probability is extremely low that Kepler would have been able to detect that planet. There are three reasons for this:

Geometry: Planets only transit their star if the orbit is aligned so as to appear edge-on as seen from Earth. For a planet orbiting a sunlike star in an earthlike orbit, this probability is about 0.5%.

Temporal: Although Kepler observed many stars for four years, there were short gaps in operations, both planned and unplanned, when observations were not conducted. Some stars were not observed in all quarters, and some earthlike planets could have orbital periods so long that four years was not enough to record the three or more transits required for detection. For a planet with the same orbital period as the Earth, the probability of three or more transits taking place while Kepler observed its star is about 90%.

Signal: Unfortunately, many stars' brightness varies over time by an amount comparable to the dimming that would be caused by an earthlike planet crossing in front of it. In these cases, Kepler could be staring right at the star while a transit occurs, but we would not be able to recognize it in the data. For an earthlike planet orbiting a sunlike star, the probability of the transit signal sufficiently exceeding its star's noise is about 6%.

The probabilities vary considerably from case to case, but they combine to leave only a small fraction of earthlike planets observable. We call the product of these three numbers the completeness, and calculate it for all combinations of orbital period, planet size, and type of star involved in this study. Completeness allows us to use a set of discoveries to estimate how many stars typically have planets of a certain kind.

Earthlike KOIs

In our sample, 59 KOIs fall into the ranges of consideration we defined above. Of those, only 8 have earthlike estimates of both size and equilibrium temperature, which consist of 5 Candidates, 3 Not Dispositioned, and none Confirmed.

In this graph, we see the 59 KOIs that are possible earthlike planets. Orbital period is on the horizontal axis and the size of the planet, in terms of Earth, on the vertical. Cases with the estimated equilibrium temperature in the earthlike range are white; those cooler are blue, and those hotter are red.

The size of the circle is based on factors that relate to how likely the planet is to exist. Larger circles are used for:

• Confirmed planets

• Candidates with

            -Five or more transits (unlikely to be evenly spaced in time by chance) or:

            -Low observational noise as defined in my post about seasonal variations in noise. This is the noise in the quarters where transits were observed divided by the noise over the three least-noisy quarters the star was observed. If this is below 1.15, the observational noise is considered low. Higher noise is associated with higher probability that the object is an electronic false positive.

Areas in the graph where completeness is greater than 0.0001 are shaded pink.

Completeness of 0.0001 on a sample of 144,308 stars translates into a frequency of about 7%, which is near the 5.7% frequency of earthlike planets around sunlike stars derived by [Petigura, et al], although the comparison is not so straightforward since this chart shows false positives and the definitions of "earthlike" are not identical. In any case, it can be seen that most earthlike KOIs fall near or into the shaded region, whereas the lower completeness region at lower right makes the detection of such worlds difficult. For illustrative purposes, the symbols for Venus and Earth have been placed at the appropriate locations.

As the graph shows, there are five cases where the estimated parameters agree exactly with the narrow definition of “earthlike”. There are many more cases where the size, temperature, or existence of the planet is particularly in question. In an almost insidious fashion, each of these qualities tends to be doubtful precisely when the others are better established, as we can see in the cluster of large white circles with periods less than 100 days and sizes of about 1.3 R⊕. These worlds appear to be the right temperature, but are larger than our cutoff of 1.25 R⊕. Just below those, we see many large red circles that indicate an earthlike size but higher temperature. Elsewhere, we see many small white circles which are the right temperature but may not exist. It is not a coincidence that earthlike size and temperature are rarely found together in Kepler results, because each of those qualities lowers the completeness of the observations, and in combination, earthlike size and temperature greatly reduce the completeness. As a result, we have at present no case which is known to exist and be earthlike in size and temperature. But there are many cases which look favorable. How many of these are real?

Expectations

If we integrate completeness as explained above across the entire parameter space, we can consider how many earthlike planets we should expect to find in Kepler data as a function of how common such worlds actually are.

If 100% of stars had one real earthlike planet, then we would expect Kepler’s data to include 4.0 of them circling class M stars, 4.2 around class K stars, 7.9 around class G stars like the Sun, and only 0.6 around class F stars, whose habitable zone is largely out at longer periods with resulting low temporal completeness.

But nobody expects 100% of stars to have earthlike planets, because even if many stars have a planet in an earthlike orbit, many of those will not earthlike in size (in our solar system, only 2 out of 8 planets are earth-sized). [Petigura, et al] estimate, by extrapolation from larger and hotter Kepler discoveries, that 5.7% of sunlike stars have an earthlike planet. As noted earlier, their definition of earthlike is different than the one I use here, but it has a similar area in parameter space, and is useful for setting approximate expectations. If that 5.7% held true for our purposes, then the K, G, and F stars on the list should have a total of 0.7 such planets, which is a tantalizing number, indicating that there is probably an earthlike planet, but maybe none, and probably not more than one. [Dressing & Charbonneau] concluded that about 15% of class M stars (red dwarfs) have earthlike planets, which would lead us to an estimate of 0.7 such earthlike planets on the KOI list, for a total of 1.4 across the four stellar classes.

Earth Two

The table below summarizes the properties of the 59 possible earthlike planets in the KOI list. The six most promising cases are coded white. Cases with an equilibrium temperature (eqt) that is too cold or too hot are coded blue or red, respectively. The remaining cases, where the size is well outside the range 0.8-1.25 R⊕, the KOI is not dispositioned, or the Observational Noise is greater than 1.15, are coded gray. The three terrestrial planets with atmospheres in our solar system are included for comparison’s sake, at the top, coded yellow.

We should expect that about 1 or 2 earthlike planets are among these KOIs, but that number is not guaranteed. Systematic or Poisson errors could make the true count as low as zero or as high as several. And if some of the worlds on this list are truly earthlike planets, we don't know which ones. The most likely possibilities are the six coded with white which I will briefly discuss as a group.

First, all six are Candidates, none Confirmed. Each is the only KOI detected in its system. This is important, because it has been estimated that a significant fraction of Candidates are actually false positives but that the FP rate is much lower for multiple-KOI systems and consequently higher for single-KOI systems. This is compatible with the estimate that approximately two of these Candidates are real.

Four of the six were observed making five or more transits, which is evidence that they are real astrophysical objects, whether astrophysical false positives or not. The other two made three transits each, and are subject to suspicion of being electronic false positives.

The uncertainties in size and equilibrium temperature, not shown, allow for any of these worlds that does exist to be outside the ranges defined as earthlike. The uncertainty in equilibrium temperature is relatively small in most cases. However, the uncertainty in size is often considerable – larger than the size estimate itself in most cases – so it is possible that any world listed here, if real, could actually be larger than 1.25 R⊕. Size and equilibrium temperature don’t pin down the climate these worlds might have. For example, Venus is, strictly speaking, earthlike as we have defined it, but in reality is famously hellish, with a surface temperature much hotter than its equilibrium temperature due to a runaway greenhouse effect. It has been theorized that planets orbiting a red dwarf star experience tidal phenomena that could make them un-earthlike in various ways. The range of climates of other terrestrial planets remains a topic for further study.

The Path Ahead

It is possible, then, that the first earthlike planet humanity will find (or has found) is on this list, but it would require more work to establish which of these Kepler discoveries qualify as such. It may be that the distance to Kepler discoveries, typically over 1,000 light years, will preclude or seriously postpone opportunities for meaningful follow-up science. The next steps may, instead, focus on other exoplanets located closer to Earth. It may also be that digging deeper into Kepler data will result in a confirmed earthlike planet that is not on this list. Whatever comes next, Kepler has given us a sense of how common earthlike planets are, and if its discoveries don’t provide our first known example, they certainly assist in planning any further studies.

Acknowledgements

Great thanks are due to Peter Plavchan and Phil Horzempa for their comments and assistance along the way.

Kepler ID	Classs	Name	Dispositionon	Period (d)d)	Size (R⊕)	eqt (K)	SNR	Noise
	G	Earth		365.3	1.00	255
	G	Venus		224.7	0.95	301
	G	Mars		687.0	0.53	207
11465869	G	K05904.01	Candidate	322.5	0.77	219	8.33	1.09
6497146	M	K03284.01	Candidate	35.2	0.93	273	12.39
8570210	G	K05545.01	Candidate	541.1	1.05	206	7.30	1.04
5091808	G	K05123.01	Candidate	288.9	1.09	265	7.24
11654039	G	K05927.01	Candidate	436.4	1.24	245	7.48	1.10
5942112	K	K05210.01	Candidate	126.0	1.24	286	6.45
8120608	M	K00571.05	Candidate	129.9	1.02	180	10.64
7416016	K	K05387.01	Candidate	297.8	1.25	167	7.38
4247991	G	K02311.01	Candidate	191.9	0.95	310	8.30
12020376	G	K05950.01	Candidate	109.4	0.99	315	7.22
11462341	K	K02124.01	Candidate	42.3	1.05	313	24.10
7619667	G	K05405.01	Candidate	103.2	1.09	314	7.11
10905746	M	K01725.01	Candidate	9.9	1.15	320	7.10
8652997	F	K05554.01	Candidate	362.2	1.16	312	7.58	1.11
7033233	K	K02339.02	Candidate	65.2	1.23	312	4.30
4172805	M	K04427.01	Candidate	147.7	1.46	164	11.02
8294683	K	K05499.01	Candidate	122.6	1.33	279	7.61
6149553	M	K01686.01	Candidate	56.9	1.33	246	5.20
5709014	F	K05194.01	Candidate	287.5	1.35	297	7.38
9292100	K	K05652.01	Candidate	91.5	1.37	302	7.38
11768142	M	K02626.01	Candidate	38.1	1.39	288	17.30
11497958	M	K01422.05	Confirmed	34.1	1.40	297	17.20
3642335	M	K03010.01	Candidate	60.9	1.41	264	18.70
9002278	K	K00701.04	Confirmed	267.3	1.46	205	13.45	1.07
9002278	K	K00701.03	Confirmed	122.4	1.54	255	47.80
8036863	G	K05465.01	Candidate	476.8	1.55	227	7.78	1.04
6106282	M	K04087.01	Candidate	101.1	1.58	201	16.82
8352009	F	K05506.01	Candidate	641.6	1.59	230	6.69	1.11
3540873	G	K04986.01	Candidate	444.1	1.60	211	6.68	1.11
8890150	M	K02650.01	Confirmed	35.0	1.28	312	15.70
9205938	F	K02162.02	Candidate	199.7	1.29	306	12.05
9674789	K	K05704.01	Candidate	96.2	1.29	313	8.20	1.08
11457664	G	K05902.01	Candidate	150.7	1.51	321	7.85
5353137	M	K03447.01	N/A	31.5	0.75	253	11.27
10579570	M	K05809.01	N/A	216.1	1.13	184	6.38
7592339	F	K05401.01	N/A	229.9	1.13	301	7.16
3641216	F	K04996.01	N/A	358.5	1.18	280	5.98	1.08
6608090	G	K05303.01	N/A	438.6	1.25	202	6.88	1.40
3865815	G	K05022.01	N/A	117.3	0.97	313	7.12
6447372	K	K05285.01	N/A	405.3	1.44	166	7.79	1.17
6364582	G	K03456.02	N/A	486.1	1.28	250	7.00	1.10
10663976	G	K05819.01	Candidate	381.4	1.29	213	7.06	1.24
4139254	G	K06108.01	N/A	485.9	1.29	226	7.26	1.05
8678345	G	K05560.01	N/A	365.0	1.31	241	7.54	1.19
10552263	G	K05806.01	Candidate	313.8	1.33	272	7.61	1.17
12645262	G	K05975.01	N/A	545.5	1.34	228	6.66	1.21
9412267	F	K05670.01	N/A	542.2	1.37	239	7.25	1.14
6946708	G	K06151.01	N/A	431.8	1.40	211	9.73	1.05
9463329	F	K05679.01	N/A	615.9	1.40	221	7.37	1.24
10977671	G	K05846.01	N/A	199.1	1.42	279	9.94
9941136	G	K05737.01	N/A	376.2	1.43	254	6.63	1.08
5529385	G	K05176.01	Candidate	215.7	1.51	292	8.33	1.40
3548044	G	K02194.03	N/A	445.2	1.55	240	9.72	1.13
11462969	G	K06239.02	N/A	491.6	1.55	221	6.61	1.05
5271637	G	K05147.01	N/A	471.4	1.55	267	8.18	1.04
12007270	G	K05948.01	Candidate	398.5	1.57	223	7.66	1.22
9024568	G	K05601.01	N/A	124.1	1.35	305	7.57
9935983	G	K05736.01	N/A	161.7	1.43	314	9.89
7041972	G	K05350.01	N/A	222.1	1.56	319	6.91