A proposal to find Earth-like planets
Planets around stars are difficult to find. Many have been found and
several search techniques exist. These are the techniques I know about:
- Doppler effect. When a
heavy planet turns around a sun it makes that sun move. The movement
causes faint shifts in the sun's color and those shifts can be measured.
- Ocultation. Rarely a
planet passes in front of its sun. That makes the sun's brightness
decrease a little.
- White dwarfs. That kind
of star lost most of its mass. Hence the planets around the star orbit
further from the star. That makes they can be detected.
- Gravitational lenses.
Sometimes a sun and one of its planets are adequately aligned with
another star and the Earth. That causes a gravitational lens effect
that allows to detect the presence of the planet.
Most of these methods detect the planet an indirect way, detect only
huge
Jupiter-like planets or depend from very special and rare events.
Better telescopes
are planed that will allow direct detection of remote planets. The
purpose of this text is to propose a method that should contribute to
detect Earth-like planets.
Let's suppose the Universe is clean and our telescopes are excellent.
What would we see looking towards a star orbited by an Earth-like
planet? We dream to see something like this, with the star on the
left and the earth-like planet as a little dot on the right:
Two things are correct in that picture:
- The distance between the star and the planet compared to the
star's diameter.
- The white and blue color tints of the star and the planet (if we
suppose the star is similar to our Sun and the planet is similar to our
home Earth).
The big error is the luminosity of the star compared to that of the
planet. Also the picture suggests the planet has a diameter of 1 pixel
while the star has a diameter of 4 pixels.
Suppose we focus on that
pixel were the planet is situated. We use a hypothetical
hyper-telescope to zoom on
it 100 times. This is what we would see:
On that picture we really can see the planet, as a 2 pixel diameter
disc. That little 100 x 100
picture is made of 9,996 black pixels and the
planet's 4 color pixels. The contribution of the planet to the
picture's global brightness is far too faint to be noticeable. If we
rescale that picture to make it back 1 pixel, that pixel will be pitch
black, the same perfect black as all other pixels in the first picture.
So in fact the planet should have been invisible on the first picture.
What's more we supposed the star and the planet have comparable surface
brightnesses. That's false too. Just compare the Sun's brightness with
the Moon's brightness when both are visible in the sky. The Moon's
surface brightness is comparable to that of Earth's. So the planet's
total brightness is tremendously weaker than that of the star.
A simple Space-based telescope would have no chance to spot a planet
because the
star's brightness will overwhelm the telescope's camera. Just have a
look at photographs of galaxies. The stars of our own galaxy present in
the picture field virtually burn away the camera's sensor in a few
pixels radius. Those stars appear quite huge, while their real diameter
is only a minute fraction of a pixel. Masking away the star will
help. Masking devices are planned for future space telescopes.
I suppose a simple Earth-based telescope has yet another problem to
spot a planet: the light produced by its star will overwhelm everything
due to light scattering in the Earth atmosphere:
Maybe use a mask in astrosynchronous orbit to help the Earth-based
telescope... Or a high altitude dirigible.
Whatever technique used to dimm away the star's light, detecting the
faint brightness of the planet will be difficult.
There is a way to detect if a pixel has a slight brightness
increase. Therefore a huge amount of photographs must be taken, all the
same as the one just above. Maybe a million of them. They must be fed
into a computer. Let's explain this with a practical example. Suppose a
team of astronomers believes a planet is situated inside a 8 x 8 pixels
square at this position around the star:
Our job is to determine if the planet is present and where exactly it
is situated inside the 8 x 8 square.
Therefor we will use photos of the 8 x 8 region.
Each photo will be strongly exposed. Say we seek an average luminosity
of 100. That yields little photos like this one:
That little photo seems all gray. In fact some pixels are slightly
brighter and some are slightly darker. That's unavoidable when taking a
photo. At random there will always be a slight difference between the
pixels even if what you photograph is perfectly uniform and your camera
is "perfect". Below is the little photo zoomed x16. If you've
good eyes and a quality display you will be able to see the brightness
of the pixels varies
slightly. Maybe help yourself by increasing your display's contrast:
This table shows the brightnesses of the 8 x 8 pixels in
numbers:
| 100 |
100 |
100 |
101 |
100 |
100 |
100 |
100 |
| 100 |
99 |
99 |
100 |
100 |
101 |
100 |
100 |
| 99 |
100 |
100 |
100 |
99 |
100 |
99 |
100 |
| 101 |
100 |
100 |
99 |
101 |
100 |
101 |
100 |
| 100 |
100 |
100 |
100 |
101 |
100 |
101 |
100 |
| 101 |
99 |
99 |
100 |
101 |
100 |
101 |
99 |
| 100 |
100 |
101 |
100 |
100 |
99 |
99 |
101 |
| 100 |
100 |
100 |
101 |
99 |
100 |
100 |
99 |
In the top row there is a pixel with a brightness of 101. This doesn't
prove at all the planet is situated there. The brightness of the planet
I simulated is of 0.01. It would never have made the brightness of a
pixel raise from 100 to 101 on its own. Besides other pixels have a
value
of 101 too. That are unavoidable errors, just like the pixels
with brightness 99. The only thing we can state is the pixel where the
planet is located will be slightly more often 101 than 99 in the many
photos.
Let's take 1,000,000 such little photos and sum them up with the
computer. We sum up
all 1,000,000 brightness values for each pixel. That yields this table:
| 100000053 |
100000393 |
100000385 |
99999508 |
100000941 |
100000470 |
100000081 |
99999820 |
| 99997783 |
100000508 |
99998497 |
100001408 |
100000061 |
99999286 |
100001270 |
100001734 |
| 99999250 |
99999024 |
99999406 |
100000668 |
100009910 |
99999918 |
99999647 |
100000261 |
| 99999297 |
100000310 |
100000249 |
99999935 |
100000042 |
99999176 |
100000227 |
99999233 |
| 100000574 |
99999072 |
100000182 |
100000313 |
100001088 |
99999329 |
100000546 |
100000926 |
| 99999716 |
100001071 |
100001168 |
99999342 |
99999512 |
100000999 |
100000133 |
100000895 |
| 99999438 |
99999494 |
100000283 |
100000695 |
100001030 |
99999697 |
99999639 |
99999414 |
| 100000159 |
99999906 |
100000208 |
100000750 |
99999914 |
99999609 |
99998716 |
100000286 |
Clearly one cell contains a value significantly higher than the others.
That's the cell in bold, the fifth one on the third row.
Let's make the table simpler by making the numbers range from 0 to 255.
That is the lowest value becomes 0 and the highest becomes 255, with
the formule
We get
this:
| 47 |
54 |
54 |
36 |
66 |
56 |
48 |
42 |
| 0 |
57 |
15 |
76 |
47 |
31 |
73 |
83 |
| 30 |
26 |
34 |
60 |
255 |
44 |
39 |
52 |
| 31 |
53 |
51 |
45 |
47 |
29 |
51 |
30 |
| 58 |
27 |
50 |
53 |
69 |
32 |
58 |
66 |
| 40 |
69 |
71 |
32 |
36 |
67 |
49 |
65 |
| 34 |
35 |
52 |
61 |
68 |
40 |
39 |
34 |
| 49 |
44 |
50 |
62 |
44 |
38 |
19 |
52 |
We can turn that table back in a picture and it yields this:
The position of the planet is clearly visible. Let's increase the size
just for the fun:
Let's ask for a cubic scaling to get the kind of picture popular
science papers like:
This method allows the measurement
of the brightness of each pixel to become very precise. So if one
pixel is just slightly brighter than its neighbors, the computer will
be able to reveal it. It will be able to tell us where the planet is
located. Once again alas this won't work. This method is excellent and
it is widely used. But in this given case it won't help. Because the
space around, in front and behind the star isn't clean at all. There
are numerous dust clouds, distant galaxies, star wind bursts... All
that makes there will be brightness variations amongst just every
pixels. We will get bright pixels smeared all over. The planets very
probably won't be amongst the brightest.
So the only solution may be to increase the telescopes' resolution.
Till we get images fine enough for the planets to emerge clearly out of
the noise. Then just a few set of pictures taken a few months apart
will allow
to be sure the dot we see is a planet orbiting that given star. Color
wavelength analysis of the dot will help too (planets have their
specific wavelength signatures).
The proposal I would like to make resembles the technique described
above. That is making millions or billions of photos and feed them to a
computer. Yet with two main differences:
- The photos have to be taken spread over a long time.
Say each second a photo is taken, during a year.
- Simply summing up these photos would be nonsense since the planet
moves around the star. Instead, all possible planet orbits around the
star must be calculated by a computer. When considering one possible
planet orbit, each photo is rotated and flattened before the pixels are
summed up. That way, should the orbit being considered be the right
one, the planet will remain at the same place on all rotated photos.
Hence it
will appear clearly on the final sum. In other words: all orbits tried
out will yield
nothing, except the one that by chance is the right one. That day a
planet is detected.
This method relies on the planets' motion to detect them. Indeed all
other sources of noise on the photos will move on different paths, so
they will cancel each other out on the final sum.
Let's illustrate this with an example. Suppose we took 31,536,000
photos during an Earth year. This is photo number one:
One of the many possible orbits for a planet around the star is shown
beneath, in blue. The white dot on the right is a possible start
position for the planet. The start position doesn't matter, I just
suppose one to make the explanation more obvious:
After 3 Earth months time the planet would be at this position, above
right of the star on the photo:
What must the computer do to add these two pictures? First it must
flatten the images to make the orbit circular:
Then the images must be rotated so the position of the fictive planet
becomes the same (whatever that position is):
When these two pictures are added, centered around the star, the
brightness of the planet adds to itself in the same pixel or set of
close pixels. Hence if we add enough different pictures we're sure the
brightness of the planet will become distinguishible. Again, there is
little chance that the one given orbit we supposed be the right one.
Far out the most probable outcome is we get nothing on the final
result. But one of the many orbits we try out will be a good
one...
At least a supercomputer will be necessary to compute out all possible
orbits, distort the images accordingly and sum them up. A high
technology telescope of today is needed to get pictures
sharp enough for the experiment. I don't know if such a telescope can
be dedicated to get millions or billions of photos of one single star
and its close surrounding.
Anyway I believe this method can yield results with today technology.
The computer does not have to compute out all possible orbits.
The computer just has to compute out all possible planet rotation
speeds around the star, whatever the distance between the star and the
planet.
Then it must compute out the ways those orbits are flattened
or expanded towards circles (because of the angle we look at the star).
Once a
planet is found, the distance at which it shows to be from the star and
the supposed rotation speed immediately will yield the star's mass.
Also at first only near-circular orbits are to be considered, like the
ones of most planets around the Sun. Only if no planet is found then
elliptical orbits can be tried out too (if funds for the supercomputer
are
still available). If the star has at least one big planet
like Jupiter,
it will be detected faster and will yield data on the star that will
boost
the finding of other fainter planets.
The mathematical method shown here is not the only one possible to
achieve the calculations. Depending on the type of supercomputer or the
mathematical skills of the researchers other methods can be used.
Simplified integer algorithms that use no picture scaling, Fourrier
Transform to get the planet's orbit as a pike on a graph... Yet the
principle remains the same.
What telescope resolution is needed to get a result? Actually this is
not a key question. Rather there is an equation between the telescope
resolution and the number of photos needed to get a result. The lower
the telescope resolution, the more pictures you need to feed the
computer. Even a telescope that confounds the star and the planet in a
same pixel could do the job, provided a reliable method exists to
center the star on the photos and millions billions billions photos are
taken and computed. For example a whole starfield can be photographed.
The star cloud as a whole can be used to center each star a reliable
way. Then the chant of the orbit of all planets around all stars can be
computed out.
Fresnel telescopes seem promising:
Eric Brasseur
-
July 5 2004 till August 31 2009
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