Cepheid Variable Stars

Velocity Graph Analysis

When the radial velocity data for a Cepheid variable star is graphed, a considerable amount of information can become available. This presentation is a suggestion of some areas of analysis.

For this discussion, the Cepheid RT Aurigae will be used, but any Cepheid where a significant number of accurate radial velocity measurements have been determined can be equally analyzed.

data1

The above graphs show both the radial velocity data and the brightness data of RT Aurigae (per Duncan). We are most specifically interested here in the radial velocity graph.

data2

This is a "smoothed curve" graph and is a best-fit to a group of around 25 data points. The radial velocity data generally has significant error factors since the values are determined entirely by rather subtle Doppler shifts in the spectral lines. Therefore, the raw data is less useful for these purposes, and a smoothed curve is used here.

Acceleration Data, or Gravitational Force Data

This graph shows that RT Aurigae varies in apparent radial velocity from about +38.5 km/sec to +4.3 km/sec, and that the curve is asymmetric. The upward sloped portion in the middle of this graph has relatively constant slope. This slope represents the part of the cycle from approximately median radius while expanding outward to the later median radius while collapsing. This is generally assumed to be a ballistic movement due exclusively to gravitational force.

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The slope of a velocity versus time graph gives the acceleration. Therefore, in that upward sloping section, there is a change of radial velocity by around +27 km/sec which occurs over a time interval of around 167,000 seconds. This indicates that there is a relatively constant acceleration which is in effect in this portion of the cycle which is +27,000/167,000 or +0.161 meter/sec2. (Plus here meaning receding from us, and therefore actually inward. Below, it is more appropriately described as negative, in the star's rest frame.)

The relative constancy of this slope suggests a relative constant accelerating force is acting during that extended portion of the velocity curve. That force is assumed to be the "surface gravity" of the star, and that the material involved is therefore following a normal ballistic path from soon after the point of having the most outward velocity to just before having the greatest inward velocity.

Since Newton's Laws show that only the mass and radius of an object are involved in establishing the surface gravity of any celestial object, this slope value seems to represent an extremely accurate value which can be used to establish either the radius or the mass if the other is known.

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It might initially seem strange that the surface gravity of RT Aurigae is only around 1/60th of Earth's gravity. After all, our Sun has a surface gravity of around 28 times Earth's. But Cepheids in general, and RT Aurigae in specific have very large radii, which causes this low surface gravity.

The actual calculations are as follows:

Gstar/GSun = Mstar * rSun2/MSun * rstar2

The current estimates for RT Aurigae are:
Mstar = 5.0 * MSun
rstar = 38 * rSun
The surface acceleration on the Sun, GSun, is around 270 m/s2

These accepted values give a surface gravitational acceleration for RT Aurigae, Gstar, of 0.93 m/sec2. This value seems certainly incorrect by a factor of about six, as the slope of the velocity curve (calculated above, around 0.16 m/sec2) must certainly represent a very reliable value for that acceleration. This seems to suggest that the currently accepted values for either the radius or mass, or both, of RT Aurigae must be incorrect. This seems to suggest some serious problems. If the mass of that giant star is actually only 1/6 the accepted value, then its total mass is less than the Sun, which seems unlikely. If the radius is 2.4 times as large as the accepted value, the star seems unreasonably large.

A possible implication from this is that the glowing gases that we sense with our spectroscopes might be well above the actual surface of the star, which could allow the acceleration value found from the velocity slope to be compatible with a relatively realistic size and mass star.

The acceleration information for the entire cycle can be found by differentiating the velocity graph by time. The resulting values for RT Aurigae from the graph above are included in a table below. The good constancy of the amplitude of that acceleration from Phase 0.080 to 0.640 seems to imply that the glowing material is following a ballistic course, where only gravitational attraction is affecting it. If there was fluid resistance of any sort, as if in a volume that also contained other interfering gases, these glowing materials would certainly attain a Terminal Velocity and cease to further accelerate. Therefore, it seems reasonable to conclude that everything in the vicinity of the glowing material is moving ballistically together. In some Cepheids that appear to have a velocity curve distortion roughly in the center of that slope, that could be an indication of a harmonic pulsing, where a collapsing shell of material momentarily encounters an outward moving shell, causing a brief deceleration of both shells.

Radial Distance Data

In the same way that differentiating the velocity graph gives acceleration information, integrating that graph gives relative position data. If a horizontal line is drawn across the velocity graph about halfway up (at around +21.6 km/sec) then the portion of the velocity graph that is below that line represents movement relatively toward us, and the portion above that line represents movement relatively away from us. The two areas involved must obviously be identical, as they represent the total distance traveled. These results for RT Aurigae are also included in a table below, which suggests that RT Aurigae alternately collapses inside a median radius by around 810,000 km, and then expands beyond it by around the same distance. The total range of radial movement is therefore around 1.62 million km. This value should be quite accurate as it is simply the integral of the (relative) velocity graph, or the area inside half the (relative) velocity graph.

The actual distance range is certainly greater than this because of the geometry of the situation. The spectrum that we receive is certainly an "average" spectrum for the full area of the star's disk that is exposed to us. The certain spherical shape of the star means that we would see a Doppler shift that is more representative of a large radiating area of the star centered on the sub-Earth point, and it is generally accepted that the actual radial velocity of the sub-Earth point would always be around 1.4 times what we measure. This implies that the total range of radial movement of RT Aurigae is probably in excess of 2.2 million km.

Finer Details

Differential Gravitation Effects

If the radius of RT Aurigae is around the 26 million km that is currently accepted, then the variation of 2.2 million km in star radius represents about an 8% change in radius. The gravitational local acceleration should therefore be around 17% weaker at the maximum radius as it is at the minimum radius. This implies that the slope of the velocity graph should be 17% less when the star is at greatest radius. Since the relatively straight section of the collapse all occurs outward from the median radius, there should be roughly half this much of a change in slope near the center of that section. An 8% reduction in the slope of the central portion of that line would therefore be expected, if the motion is truly purely gravitational and the mass and dimensions of the star are as they are believed to be. The data presented here is not extensive enough to identify such an 8% reduction in slope in that central portion unambiguously. Clearly, further accurate velocity measurements could be of great value here.

Differential Driving Accelerations, Forces

Remembering that this particular data set is very limited, it is still intriguing that the outward acceleration amplitude (between around phase 0.740 and 1.000) appears to include an interesting characteristic. The acceleration is not constant, as it might be with a standard spring constant. Instead, it appears to increase as the effective radius of the star decreases. It could be extremely informative if extensive accurate data is accumulated in this portion of the cycle. If it should be later determined that the outward acceleration varies as a second power, or third power of the radius dimension, which would imply that the driving force does likewise, some valuable insights into the actual nature of the driving force might be developed.

In addition, if that third derivative curve can be accurately developed for that portion of the cycle, and therefore accurate information regarding the dependency of necessary "rebounding force" on radius, then an additional dimension might be developed. A specific radius might be identified (inside the star) as being critical to defining the pulsation process, which would add additional information to our knowledge and theories regarding Cepheids. If the meager data from this sample could be relied on, then the rebounding / repulsive force, which was likely around zero at the median radius, had gotten to six times the local gravitational force, one to counteract that gravitation and five more to create the acceleration. This great change happened in around 806,000 km of radius difference. It is suggested that an accurate data plot might be extrapolated to a "critical radius" that would represent an apparent infinite repulsive force. This data seems to suggest that such a radius would not be very deep inside RT Aurigae. Further analysis in this direction was not done, due to the small number of data points in this sample.

Data on RT Aurigae derived from the Duncan Graph Above

PhaseVelocityRel. VelocityAccelerationDistance
km/seckm/secm/s/skm
0.000 +4.22-17.38 +0.1153 -310,000
0.040 +4.84-16.76 -0.0476 -90,000
0.080 +6.39-15.21 -0.1208 +116,000
0.120 +8.27-13.34 -0.1453 +300,000
0.160+10.24-11.36 -0.1534 +459,000
0.200+12.74 -8.86 -0.1941 +589,000
0.240+14.93 -6.67 -0.1697 +689,000
0.280+17.22 -4.38 -0.1778 +760,000
0.320+19.62 -1.98 -0.1860 +801,000
0.360+21.70 +0.10 -0.1615 +814,000
0.400+23.99 +2.39 -0.1778 +798,000
0.440+26.07 +4.47 -0.1615 +753,000
0.480+28.26 +6.66 -0.1697 +682,000
0.520+30.44 +8.84 -0.1697 +582,000
0.560+32.52+10.92 -0.1615 +455,000
0.600+34.61+13.00 -0.1615 +300,000
0.640+36.48+14.88 -0.1453 +121,000
0.680+37.82+16.22 -0.1045 -80,000
0.720+38.54+16.94 -0.0557 -293,000
0.760+37.69+16.08 +0.0664 -506,000
0.800+33.89+12.29 +0.2944 -689,000
0.840+26.74 +5.14 +0.5550 -801,000
0.880+17.18 -4.42 +0.7422 -806,000
0.920 +9.92-11.68 +0.5631 -702,000
0.960 +5.71-15.89 +0.3270 -524,000
1.000 +4.22-17.38 +0.1153 -310,000
NOTES:
Around Phase 0.360 (36% of the way through the cycle beginning at the moment of greatest brightness), the Center-of-mass Relative Velocity changes from negative (approaching us, or expanding) to positive (receding from us, or collapsing). That indicates the situation of greatest radius. At that moment, the Acceleration is negative (inwardly acting, assumed gravitational). At that moment also, the radius of the star is roughly 814,000 km greater than its median radius.

After Phase 0.840, the Relative Velocity changes from positive to negative, which indicates the end of the collapse, at a star radius a little more than 806,000 km smaller than its median radius. at this time, the outward acting acceleration reaches its peak amplitude, around 5 times as large as the local gravitational acceleration would be.

In the (Measured) Radial Velocity column and Relative Radial Velocity column, plus numbers mean moving away from Earth. The opposite is true in the Acceleration column, where negative values indicate acceleration away from us, toward the center of the star, or a gravitational collapsing effect. The (Measured) Radial Velocity column is the value calculated from the Doppler shift of the lines of the spectrum. The (Center-of-mass) Relative Radial Velocity column is the value after the constant receding velocity (21.60 km/sec) of the star is eliminated, and is therefore the actual relative radial velocity of the surface, from the reference frame of the center of the star. In the distance column, the values indicate a differential radius of the star, so negative values represent a smaller star and positive values represent an expanded star.


This presentation was first placed on the Internet in February 2003.

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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago