Amateur Astronomy on Mars?

 

  The country is currently facing automatic budget cuts in January known as sequestration if Congress doesn't come up with some stopgap measures. This means that public support for the Mars programs will be cut proportionally. This will not decrease the functionality of the Mars rovers but their controllers might feel the pinch. One possiblity that NASA might consider is more private support and some "amateur work."

 

  As an example Curiosity might try some amateur astronomy with the M-100 Mastcam. It has a focal length of 100mm with an f/10 lens and can capture 1600x1200 6.8°x5.1° images. So, time permitting, it might be possible for Curiosity to observe the location of the Martian rotational pole. In Curiosity's case it would probably have to be the South Celestial Pole. A complicating factor is that Curiosity is in a depression about 5° S of the equator so the pole would be very close to the horizon in a dusty environment. But the visible star trails that result from the rotational motion of Mars might still be useful in determing the location of the South pole even if it is not clearly visible. The S pole is diagonally opposite the N pole so it is at RA 9h10m44s, Decl -52°53'11.4" and there are some bright stars nearby in the constellation Vela that could be used to triangulate the pole's position. In the image below one can see the marked position of the S pole and the surrounding stars of Vela.

 

 

North Celestial Pole 03b

 

  I again used the rotation center formula to calculate the pixel position of the center of rotation in the image using the pairs of points within 1000 pixels of the approximate center. The root mear square error for the pair distances from the center was 1.59 pixels. The RA and Decl are known for α UMi and λ UMi and they can be used to obtain a scale factor which can be used to convert pixel separations to angular distances.

 

 

  Letting e be a unit vector representing the position of a star and using the formula equating the dot product of two positions to the cosine of the angle between them we get two conditions for direction of the pole. Each condition specifies a circle on the celestial sphere with the known angular distance of the pole from the star. We can represent the circle about λ UMi by an equation for a circle on the sphere.

 

 

  The second condition tells us that the position of the pole is a zero of a nonlinear equation which can be plotted to estimate the angles for the zeros.

 

 

  Substituting the equation for the circle from the first condition into the second condition we get a simple equation which is linear in terms of the sine and cosine of an angle that can be solved for an expression for the sine of the unknown angle.

 

 

  Plugging the constants into the expression gives the sine of the angle, the angle and the direction of the pole in Cartesian coordinates.

 

 

  We can now calculate the Right Ascension and Declination of the computed pole in the J2000 coordinate system.

 

 

  Twelve years have elapsed since the year 2000 and we can expect some movement of the pole due to precession and nutation. The observed pole is about 9 arc minutes from the J2000 pole. The expected error in the observed pole is about 14 arc seconds.

 

North Celestial Pole 3a

 

 

 

  I zoomed in on the pole a little more Saturday evening and was able to see more of the fainter stars near the pole. The bright star at the top of the image is Polaris, α UMi, and the brighter star towards the bottom center is λ UMi. If you look slightly above and to the left of Polaris you will see a small star and a much smaller companion which is an optical binary. Some of the stars just visible are approximately 9th magnitude.

 

  The second image is a merger of 6 images taken approximately 8 minutes apart. With greater magnification one has to be more careful about camera motion. If you look carefully at the rotational track you will see that one of the stars is out of alignment. It was not used to determine the center of rotation which is marked with a +.

 

  These images are approaching the limit of what one can do with a digital camera with 26X optical zoom. Magnification reduces the speed of the lens so longer exposure times or a larger aperature is needed. One could probably improve on the position of the pole by using zooming in more with a good telescope.

 

Distant Suns Star Chart

 

 

  The above image is approximately the same field of view in Distant Suns as the images taken on Wed evening. The white lines mark the angles of Right Ascention corresponding to 0h at the top and 6h, 12h and 18h going clockwise. Part of the handle of the Little Dipper is also marked. The closest star on the handle is δ UMi (mag. 4.35)and the one just visible at the edge of the image is ε UMi. (mag. 4.21). All the stars visible in this star chart are greater than magnitude 7. To give a sense of scale ε UMi is about 8° from the pole.

 

 

North Celestial Pole 2

 

 

  On Wednesday evening I captured several images of the North Celestial Pole like the one above. Each image was 4288 x 3216 and the exposure was 16 seconds with an F-Stop of 4.5. Six of the images taken about 8 minutes apart were merged and the ends of the arcs were used to calculate the pole using the formula for the rotational center. My results agreed fairly well with the J2000 position. The humidity was about 75% and there was some sky glow present. The brightest star near the center is Polaris.

 

 

  Supplemental (Nov 22): I've replaced the original merged image which used the sum of the pixel values of the image points with one that uses the greater of the two pixel values. There's a little more background noise but the pixel values are more representative of the observations. The images posted are one sixth the size of the originals. MS Paint was used to get the pixel coordinates for the data. The "+" marks the computed pole position.

 

Locating the North Celestial Pole

 

img src: Wikipedia

 

  The formula used to find the center of a circle can be used for other applications as well. The formula computes a position that will minimize the difference between pairs of points. The radial distance from the center does not have to be the same for each pair. This will enable us to calculate the position of the North Celestial Pole on an image using the endpoints of a streak formed by the motion of a star. The calculation of the center of rotation is shown below.

 

 

Checking Event Times With Mars24

 

  One can view the position of Mars for a specific date and time on the Mars24 plots by using the Settings tool found in the Window drop down menu. The orbit and analemma plots for perihelion are shown below. L_s in the analemma plot appears to be the Sun's RA given in degrees of longitude and is used in place of a date.

 

 

 

  Supplemental (Nov 19): The Mars24 technical notes state that L_s is the areocentric longitude of the Sun measured from the Martian equinox. This is probably the relative longitude along the celestial great circle of the Martian ecliptic due to the Sun's motion in its "areocentric orbit." L_s is also used in connection with the ephemeris time (ET).

 

Nov 15th MSL Teleconference

 

src: nasajpl

 

  Today's telecon dealt primarily with Curiosity's local environmental conditions such as dust devils, pressure fluctuations and radiation levels. The pressure fluctuations not only exhibit daily changes but also changes over a longer timeframe. Mars is currently moving closer to the Sun and will reach perhelion on Jan. 23, 2013*. The radiation on Mars is more easily affected by its environment in space since it has lost its magnetic field and has a relatively thin atmosphere as compared with Earth conditions. The radiation levels fluctuate during the day and are greatest when the atmospheric pressure is lowest. The local winds also fluctuate throughout the day and move up slope on Mt Sharp and Gale Crater's rim during the day and down slope at night. Curiosity is near the center of these two flows and is in the region where winds flow around Mt Sharp.   related press release   telecon images   Timekeeping on Mars   Mars24 Sunclock

 

*Supplemental (Nov 15): MICA's perihelion is 2013 Jan 24 08:56 UT

 

MSL on NOVA

 

Watch Ultimate Mars Challenge on PBS. See more from NOVA.

 

  Some background on the development of the Mars Science Laboratory rover Curiosity was shown on the PBS program NOVA Wednesday evening.

 

Total Solar Eclipse

 

 

  There will be a total solar eclipse on Tues. Nov 13, 2012. Unfortunately the only major landmass for which totality will be visible will be the northernmost part of Australia. The eclipse will start at 19:38 UT and end at 0:46 UT. see details

 

Estimated Circle Center & Radial Residuals

 

  I may have violated my position on avoiding misnomers in the last blog. The residuals there would more correctly called radial pair differences. With an estimated center for the circle one can compute a mean radius for the initial points and a better set of radial residuals as well as their rms error. The expected error in the mean radius would be ε_rms/√5=0.173. Half the radial errors would have a magnitude less than 0.512 pixels as might be expected when working with integers.

 

 

More on the Least Squares Estimate of a Circle's Center

 

  The formula for the least squares estimate for the center of a circle that was used in a recent blog uses the difference in the square of the distance from the center for pairs of points as the difference whose square is to be optimized. One can study how the formula works for different sets of pairs of points. A path can be drawn between the original five points that will link all the 10 pairs of points in a sequence. The first two pair of points with the largest separation gives the center of the circumscribed circle through the set of three points. The same is true for the three pair pairs of points for the triangle with the largest sides. As one adds more pairs of points the estimated center is less dependent on the individual points used and tends to converge but there is some residual difference between the distance of a point from the center for the pairs. Squares of the distance of a point from the center were used to compute the differences to avoid the use of square roots in deriving the formula. The method is similar to using hyperbolic navigation to determine one's position.

 

 

 

 

 

 

  Supplemental (Nov 2): I got residuals that differed in the third decimal place from the radial residuals for the 10 pair of points above using a method for determining position based on differences in signal arrival times. The data points for the circle were treated as signal sources whose positions were known and the differences in arrival times were set to zero. The estimated position of the center also differed in the third decimal place.

 

Nov 2nd MSL Telecon: Negative Result for Methane on Mars

 

src: nasajpl

 

  Today's MSL teleconference focused on the SAM suite of instruments used to study the composition of gases in the Martian atmosphere and the presence of volatile compounds from the surface. The Tunable Laser Spectrometer is can measure the isotopic composition of the carbon compounds in the atmosphere. The relative abundance of the isotopes present can give information on the escape of gases from the Martian atmosphere over time. Heavier isotopes are less likely to escape which results in a higher abundance. The instument was not able to detect the presence of methane in the atmosphere. The conclusion is that if there is any methane present the sinks on the surface dominate over the sources.  telecon images  related press release

 

MSL's 1st X-ray Diffraction Pattern (1D)

 

  I tried to convert MSL's first 2D diffraction pattern to a 1D pattern. The first problem is to find the center of the pattern of rings. This was done by doing a least squares fit for five pixel points on one of the rings. The calculation goes as follows.

 

 

  One can check the fit by plotting a circle on the 2D diffraction pattern.

 

 

  A 1D diffraction pattern was obtained by measuring the distances of the image pixels from the center of the rings and summing the gray scale image values for the radii in one pixel intervals and then dividing by the number of pixels in each interval to get average values.

 

 

  The peaks in the 1D plot can be compared with powder diffraction databases to determine the minerals present. For more information on XRD and CheMin see the MSL Science Corner and David Blake's Historical Perspective.