Sunday, August 13, 2017

The Sun's Apparent Motion in the Plane of the Ecliptic

  I redid the series of fits for the Sun's apparent position this time in the plane of the Ecliptic. The primary motion is of course a Keplerian ellipse. The horizontal and vertical axes are the major and minor axes and at the beginning of the year the Sun in near perigee on the right and moves upwards. The units are AU.

The Keplerian elements for the fit are as follows.

The residuals of this fit form a rose curve which appears to be due to solar pulls and torques acting on the Moon's orbit. Again the units are AU.

The residuals of the second fit are down to μAUs and more random in appearance. There's an odd step in the direction of increasing perigee at the end of the year. Could the Earth be slowing down and spending more time at perihelion? What effect would that have on global warming?

Fits can produce some deviations when all the error isn't accounted for.

Wednesday, August 9, 2017

Found an Odd Error for a Fit of the Sun's Position

  I was preparing for the solar eclipse later this month, doing a fit of the Sun's relative position from the center of the Earth, and the fit didn't turn out as I expected. A linear least squares fit of the Sun's position for the functions indicated resulted in the following relative differences between the fit and the calculated positions.

The sinusoidal function is mainly due to the Moon's pull on the Earth but there appears to be another component present. How can one explain the displacement of mean error from zero? It turns out there are some Chebyshev polynomials present.

This polynomial series generates a displacement of the following form given in AU.

After subtracting this and the displacement due to the pull of the Moon the following error remains.

The error can be measured in micro AU (μAU). For comparison the Earth moves about the Sun at a mean rate of 2π/365.24=0.0172AU/day=12μAU/min. It probably wouldn't hurt to get accurate measurements of the start and end eclipse times.

Essentially the same error or perhaps correction is present in both MICA and HORIZONS data for the Sun's position relative to the Earth.

Wednesday, August 2, 2017

With Light Sail Acceleration in the Direction of Sunlight Preferred

  One can study various modes of light sail operation to see how it will perform. The parasitic reduction of perigee seems to be associated with acceleration in the direction of sunlight and requires more energy for insertion into a higher circular orbit. The following mode of operation prefers acceleration in the direction of sunlight. The attitude of the sail is directed as follows.

The relative change in the apogee of the sail increases with time but the perigee also decreases even if a small 10 second step is used in the calculation.

Again, more Δv is required to put the light sail into a circular orbit than for a ballistic transfer orbit. More time would be required to compensate for the loses when the Δv is in the proper direction orbit insertion.

If nature favors some modes of light sail operation over others the preferred mode would make the desired orbital changes in the least possible time. It may require a little R & D to determine the optimal solution. The rate of the gain in altitude at apogee is greatest for α=0 when the light sail moves in the direction of sunlight. If the drop in perigee isn't compensated for light sail won't climb out of the Earth's gravity well before it encounters atmospheric drag. So one will have to do the climb stepping from one circular orbit to a higher one.

Tuesday, August 1, 2017

The Effect of Changing the Step Size on the Light Sail's Orbits

  Since the light sail's orbit appeared to show signs of a deteriorating perigee I tried playing with the attitude of the sail relative to the direction of sunlight to get relatively more angular acceleration. In the figure below the direction on sunlight is to the left. The light sail's acceleration is along its normal n which is at an angle α relative to the horizontal axis. The changes to the equations of motion are also shown.

The angle of the sail was set to α=π/2-θ when it was above the x-axis and moving away from the Sun and α=π/2 when it was below in order to restrict the acceleration some. The behavior was sensitive to the size of the step so Δt was set to 10 seconds. This gave better results for the behavior of Δr=r-r0.

It still drifts away from a ballistic object in the original circular orbit due to changes in the eccentricity and the period of the orbit.

Changing the step size resulted in a closer match of the light sail's Δv's with the transfer orbit  Δv's for insertion into the higher circular orbit.

So it appears that a light sail can make changes to a transfer orbit. Unfortunate, the apogee is on the sunward half of the orbit and one can't use sunlight to accelerate towards the sun to go into a higher circular orbit. This is where an auxiliary propulsion system would be needed if one wanted to do orbit changes. A flyby mission would not require orbit insertion.

Supplemental (Aug 1): After the passage of half a year the sunlight will be in the opposite direction (to the right in the first figure) and the Δv will be directed properly for insertion into the higher circular orbit. It may be possible for a light sail to slowly work its way out of a gravity well.

Monday, July 31, 2017

LightSail 2 Needs to Compensate for the Decrease in Perigee Periodically

  One can compare the energy change in LightSail 2's orbit that with that required to go into a transfer orbit. After a number of revolutions however the perigee of the light sail starts to drop. This loss of energy has to be compensated for if one desires to go into a circular orbit at the light sail's apogee.

In the plot above the blue curve indicates the velocity change Δv required to go from a point of the light sail's orbit into a circular orbit. The solid red line is the total Δv required to go from the original circular orbit into a circular orbit at the apogee of the transfer orbit. The dashed red line indicates just the Δv required to go from the apogee of the transfer orbit into a circular orbit. Initially the orbit injection Δv's of the light sail match up with the Δv's for the transfer orbit. After about six revolutions more Δv is required to compensate for the drop in perigee. The implication appears to be that the orbit of the light sail needs to be periodically corrected to a circular orbit to keep the light sail from becoming parasitic. The major deficiency of the light sail is that the direction of its thrust is limited. It's lack of angular acceleration requires an auxiliary propulsion system for some orbit changes.

Supplemental (Aug 1): The deviations in the Δv's for the light sail when going to the higher circular orbit are affected by the step size. Here the step size was Δt=60 seconds. Compare next blog.

Sunday, July 30, 2017

How Long Would It Take LightSail 2 To Escape From the Earth?

 One can compute LightSail 2's gain in energy over time and estimate the average rate at which it will gain energy.

This average rate appears to be linear and allows one to estimate how long it would take for the light sail to acquire enough energy to escape from the Earth's gravity well. The answer is 6.4 years.

The model used was overly simplified neglecting the Earth's shadow and didn't take into account the need to raise the height of perigee to avoid atmospheric drag. As the Earth moves about the Sun in its orbit the direction of sunlight will slowly change.

Friday, July 28, 2017

LightSail 2's Behavior in Orbit

  The Planetary Society's LightSail 2 is to go into orbit aboard SpaceX's Falcon Heavy later this year if all goes well. The initial orbit will be a circular orbit 720 km above the Earth's surface. The light sail will face the Sun as it moves away from it and the plane of the sail will lie along the direction of a line from the Sun as it moves toward it. The effect of switching back and forth between its boost and cruise phases will be a slow but steady increase in the satellite's orbital energy.

The following figure defines the plane of the orbit with the x-axis pointing towards the Sun. The unit vector n indicates the direction of the acceleration caused by the light pressure on the sail during the boost phase. The equations below indicate the specific radial and angular forces acting on light sail as it orbits the Earth. μ is constant GM for the Earth.

The set of equations above are difficult to solve analytically but it's not too difficult to do a numerical calculation. The following calculations used 1 second steps in time. While in cruise phase the satellite coasts in a Keplerian orbit. The following table gives values for the selected points of the first orbit. The units for time, angle and radius are seconds, radians and meters. The highest and lowest points of the orbit are the apogee ra and perigee rp. The cruise phase was allowed to continue past the switching point at θ=2π in order to get the values for the perigee. Interpolation was used to get a better estimate of the values.

From this table we can compute some of the orbital elements for the Keplerian orbit of the cruise phase. There is a gain in energy during the boost phase.

The light sail will spiral away from a ballistic object in the same initial circular orbit since their angular separation will increase over time in addition to the radial separation.

The separation viewed from the perspective of revolutions shows the alternating crossing of the axes over time.

Supplemental (Jul 28): Two additional plots to clarify relative positions. The first is Δr=r-r0 where r0 is the radius of the circular orbit.

The second is the angular separation of the light sail from a ballistic object in the original circular orbit.

Edit (Jul 28): Found an error in the first two plots and removed them.

Edit (Jul 28): Found an error in the Δθ plot. It was a dumb mistake. You can't subtract radians from degrees. Replaced plot. Found a minor error in the spiral plot calculation which didn't affect the result much. Left plot as it was. I used 1 minute steps for the longer time period used in all the plots so there is a little additional error present.

Thursday, July 13, 2017

Refractive Index of Air as a Function of Wavelength

  In 1908 Rentschler published a study on the refractive index of a gas for different wavelengths in the Astrophysical Journal. I used his data for air in Table II to find the coefficients of a Cauchy formula to fit the data. One can also use the NIST calculator to get data for air and do another fit.

The Rentschler data appears to be for dry air. If one computes the index of refraction for the CRC Handbook data using the given wavelengths for air and computed wavelengths for the Balmer series for hydrogen the plot appears to be displaced somewhat. Note that the humidity of the air also affects the index of refraction. I used the ratio of the wavelength in air to 5000Å for the formula so it would be easier to substitute wavelengths given in nanometers (nm) using 500/λ with the same coefficients.

Monday, July 10, 2017

Balmer Series Fit Using More Accurate CRC Data

  The hydrogen wavelengths in air found in the CRC Handbook of Chemistry & Physics are lines of the Balmer series and I got better results for a fit of these lines.

One has to use the reduced mass of the electron to get RH=R/(1+me/mp) where me and mp are the electron and proton masses to compute the limit of the series in air and divide that by the index of refraction for air to get the limit of the series for air.

The index of refraction for air is a function of wavelength and that also affects the observed values and so the fit has an error that varies with the number of the series.

Sunday, July 9, 2017

Scaling Factors That Could Potentially Affect Huggins' Hydrogen Lines

  Relative to the minimum deflection wavelength for Huggins' spectroscope there appears to be a progressive shift to higher wavenumbers which is a blue shift. But relative to the limit of the series, λ, there is a redshift. There are a number of factors which might cause a scale change in the observed wavelengths and need to be taken into consideration.

Calibration of the Spectroscope
minimum deflection
calibration curve

Air Wavelengths vs Vacuum Wavelengths
Definition of standard conditions
Snell's Law (measurement in air results in a blue shift relative to a vacuum )
Rydberg Constant (R is for a vacuum)

Light nucleus
RM needed for light nucleus (R is for a heavy nucleus)
reduced mass
electron mass

Doppler shift

Gravitational redshift

All of the above contribute to a change in scale. What did Huggins miss?

Friday, July 7, 2017

A More General LS Formula for Estimating a Common Factor

  One can generalize the formula in the last post to find the best estimate of a common factor.

Using this formula gives a better estimate for the wavenumber limit of the Balmer series as the following calculation shows. The formula gives 2744.8 while the average for the individual estimates is 2744.6 and the rms errors are 1.14 and 1.13 respectively.

The improvement over an average is marginal and doesn't appear to compensate for a scaling error.

Supplemental (Jul 8): Huggins adjusted his spectroscope for minimum deviation for the H line corresponding to 4340 Å or n=5. This looks like a better fit for the wavenumbers with an rms err of 0.46 which could result from rounding to the nearest Ångström.

This rescaling brings the estimate for the limit of the wavelengths to within a quarter of an Ångström of the currently accepted value.

Supplemental (Jul 8): The deflection of light by a prism depends on the index of refraction n and the wavelength λ. Shorter wavelengths are bent more than longer ones so blue light is bent more than red. The deviation from linearity also increases as the wavelength decreases so this might explain the need to rescale the spectroscope readings. The changes are in the right direction but I haven't compare the observed errors with calculated errors.

An Error in Huggins' Hydrogen Wavelengths?

  There appears to be a systematic error in the hydrogen wavelengths in Huggins' paper when one compares them with the fit for the Balmer series. Notice that there is an progressive increase in the value of the error, Δν. If one uses the first line as the reference we can set that error equal to zero and compute the relative errors Δν'.

Trying to fit a straight line to this data is a little tricky but least squares allows us to derive a formula for the best fit for a line going through the origin.

The adjusted lines give a closer estimate for the limit of the lines of the Balmer series.

It's possible that the Iceland spar prism that Huggins used was slightly nonlinear in its dispersion of light. Dispersion is dependent on frequency. This may be why rescaling gives a better result. The least squares derivation of the formula above is fairly simple. The quantities on the right of the formula are the column vectors in the second table.

Wednesday, July 5, 2017

The Balmer Series

  It's customary on the 4th of July for Americans view local fireworks displays. What the experts can do with patterns, colors and sound is quite amazing. History tells us that fireworks originated in ancient China and the ancient alchemists experimented with colors. Knowledge of the colors that can be produced can be acquired through the use of flame tests. This tool is still used in modern chemistry.

  The scientific study of colored light was advanced in the late 17th Century by Newton's work on the prism. At the beginning of the 19th Century the introduction of the spectroscope allowed scientists to study the Sun's spectrum and discover the dark lines known as Fraunhofer lines. At the same time flame tests of various elements allowed scientists to connect these lines to chemical elements present. In 1868 Ångström published accurate values for the lines of the solar spectrum with the elements associated with them.

In 1885 Balmer published a notice giving the formula for a series of hydrogen lines. How might he have accomplished this? His data came from a notice by Huggins on the hydrogen lines present in the spectra of certain stars. The data is included in a footnote referring to a note he received from Johnstone Stoney, a fellow of the Royal Society, who states that the lines might belong to a series.

What happens if we try to do an empirical fit for the data? Notice that Stoney also includes the wave numbers, ν=1/λ, and we can try to fit these. The lines appear to converge in one direction so we might first try to fit a formula that is quadratic in 1/n (fit1). The results are quite good with an rms err of 0.4. Using n=3 for the first line gives the best fit. The value for B is relatively quite small when compared with the others and the ratio of C to A is very close to 4. Redoing the fit for just two terms makes the ratio even closer to 4 so we can just try to get a value for A by computing a value for each line and taking the average.

Although the rms error is greater, we still get a fairly good fit to the data.

Notice that Stoney includes a curve passing through the data points. Did he know the formula for the series of lines? His table suggests he used a difference formula to fit the data.

Supplemental (Jul 5): Fraunhofer's lines:

Huggins Plate 33 showing the line spectra of a number of stars. The second row appears to be α Lyræ (Vega) containing the first twelve lines of the data above:

Thursday, June 1, 2017

Newton's Temperature Scale

  The thermoscope, a bulb containing air with a long tube that was immersed in water, was developed by Galileo and others to measure temperature during the first half of the 17th Century. Boyle studied similar "weather-glasses" and introduced the hermetically sealed thermometer in England by 1665. In 1701 Newton anonymously published an article, Scala graduum caloris, which described a temperature scale ranging from the freezing point of water to that of a fire hot enough to make iron glow. An English translation of Newton's article can be found in Magie, A Source Book in Physics, p. 225.

Newton's temperature scale has a geometric series and an arithmetic series associated with it. The geometric series corresponds to the temperatures and the arithmetic series is associated with cooling times.

 "This table was constructed by the help of a thermometer and of heated iron. With the thermometer I found the measure of all the heats up to that at which lead melts and by the hot iron I found the measure of the other heats. For the heat which the hot iron communicates in a given time to cold bodies which are near it, that is, the heat which the iron loses in a given time, is proportional to the whole heat of the iron. And so, if the times of cooling are taken equal, the heats will be in a geometrical progression and consequently can easily be found with a table of logarithms."

After finding a number of temperatures with the aid of a thermometer, Newton describes how the hot iron was used.

"...I heated a large enough block of iron until it was glowing and taking it from the fire with a forceps while it was glowing I placed it at once in a cold place where the wind was constantly blowing; and placing on it little pieces of various metals and other liquefiable bodies, I noted the times of cooling until all these bodies lost their fluidity and hardened, and until the heat of the iron became equal to the heat of the human body. Then by assuming that the excess of the heat of the iron and of the hardening bodies above the heat of the atmosphere, found by the thermometer, were in geometrical progression when the times were in arithmetical progression, all heats were determined."

Newton's temperature scale can be constructed mathematically as follows where I've noted some corresponding temperatures on the Fahrenheit temperature scale for comparison.

The temperature point between the melting point of wax and the boiling point of water is an average. I used the geometric average which works best. One can put together a table as follows to compare the Fahrenheit temperatures with the index number, k, above.

A graphical comparison shows that the logs are fairly linear. Using 66°F for the temperature difference gave the best fit for human body temperature at the lower left of the plot.

The slope of the fitted line can be used to convert Farenheit temperatures to points on Newton's scale.

Newton's law of cooling can be in be expressed as the difference between the temperature of an object at some time and the ambient temperature being proportional to an exponential term involving time. This can to shown to be equivalent to the differential form of the law.

Supplemental (Jun 1): Leurechon Thermometer (1627)

Supplemental (Jun 2): 65°F gives a better fit for body temperature. Was this the ambient temperature at which the experiments were done? It's doubtful there was a standard temperature yet in Newton's time. For more on the history of early thermometers see Bolton, Evolution of the Thermometer, 1592-1743.

Supplemental (Jun 2): The average of the freezing point of water and body temperature is (32+98.6)/2= 65.3. Did this originate with Accademia del Cimento?

Supplemental (Jun 4): Corrected conversion formula for k.

Monday, May 22, 2017

Fermat's Problem in Three Dimensions

  Verified the Newton's method works in three dimensions. I choose the  four vertices of a tetrahedron as the given points. The Fermat point which makes the sum of the distances from the given point a minimum turned out to be mean of the given points.

I used Excel to create an anaglyph. You will need red-cyan glasses to view it properly.