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.

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.

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 r_a and perigee r_p. 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.

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.

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 R_H=R_∞/(1+m_e/m_p) where m_e and m_p 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.

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
R_M 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?

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.

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: