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Roxy's Ruler

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page: Roxy's Ruler timestamp:20190310194416 year: 2019 month: March day: 10 substed: yes

Roxy's Ruler is a name for a method of measuring the distances to spiral galaxies using spiral morphology [1]:456,480 and the galaxy's rotation curve[2] or line width.

The spiral shape of a galaxy having a flat tangential velocity of 152 kps. The scale is in thousands of light years giving the size of the galaxy from which its distance can be found.


The morphological parameters of spiral galaxies can be directly measured from photographic plates[3][4] and both line width and rotation curve can be measured in many different frequencies using radio astronomy[1]. This method has the advantages of being independent of red shift and luminosity. If vmax is the maximum orbital speed of material in a galaxy in kilometers per second (kps)[2] and αs is the measured distance between spiral arms along the major axis of the galaxy, then the distance to the galaxy, D, in parsecs, is given by the formula[5]:

D = 3.12 X 109 / (vmax X αs)

A demonstration of how to use this method can be found in WikiHow.

Cepheid distance vs.Roxy's Ruler showing a discrepancy from matching a one to one linear fit by 0.016. The confidence variable is 0.9104.

A Brief History of Galactic Distance Measurements[edit]

An acknowledgement that various nebulae lay far beyond the bounds of our own galaxy did not occur until 1920, as a result of the great debate[6]. Since then research into galactic structure and distances to galaxies exploded into a plethora of both amateur and academic papers and study.


Shortly after the realisation that the universe is much larger than previously thought, an already discovered method of measuring the distances to Cepheid variable class stars[7] was utilised to find the distances to relatively nearby galaxies by Edwin Hubble. Hubble compared the distances to these galaxies to their red shifts[8] and concluded that the universe was expanding with led to the theory of the Big Bang. This method of measuring the distances to galaxies extends to about 22 MegaParsecs(MPc)[9].


Later it was discovered by astronomers R. Brent Tully and J. Richard Fisher[10] and published in 1977, that there existed an empirical relationship relationship between absolute luminosity and the maximum rotation velocity of a galaxy, vrot. The parameter, vrot, is determined through line width. Line width is also known as the asymptotic rotation velocity of a galaxy. The Tulley-Fisher relationship states that the luminosity is relative to vrot squared. This discovery increased the ability to determine the distances to galaxies independently of red shift and advanced understandings of the large scale structure of the universe.

Comparison graph between measures using Roxy's Ruler vs Tully-Fisher. The fitted line passes through the origin and has a slope of 1.03 and the fit yields a sigma of 1.38 Mpc.

Cosmic Distance Ladder[edit]

Various other means of measuring the distances to galaxies have been found and are grouped together in various rungs on a ladder measuring farther and farther into space. This is known as the cosmic distance ladder.

Discovering Roxy's Ruler[edit]

Plot of observed rotation velocity (B) compared with rotation curve predicted (A) from assuming a Keplerian model.

The flat velocity rotation curve of NGC3198, as measured by K.G. Begeman, indicated there may exist some form of non-luminous material, i.e. dark matter, causing this effect. A three-line algebraic calculation [11] showed that a flat velocity rotation curve of NGC 3198 could only be explained if the galaxy was a linear orientation of material having a constant linear density according to Newtonian orbital dynamics. As bizarre as this result seems, it also complied with measurements of luminosity distribution made by Harlow Shapely[citation needed]. A successful determination of the algebraic formula for a spiral galaxy was found to be:

ř = θ / ῶ0.

Here ř is distance in light years (ly) and 0 is in radians per year and is found from the formulae:

0 = ṽmax / (2π)


max = vmax /c, where c is the speed of light.

This is the formula for an Archemede's spiral[12].

By overlaying the spiral produced by the above equation onto a photograph of a corresponding spiral galaxy, the intrinsic size of the galaxy is known and its distance determined. Roxy's Ruler has been used to explain the structure of spiral galaxies[13], galactic rotation curves[14] and to argue for a static universe[15].


There are significant consequences to the formula known as "Roxy's Ruler" and its derivation. No only can the distance to the galaxy be determined, but so can its mass, linear density and linear momentum. Furthermore, the Tulley-Fisher relationship is found to have a mathematical foundation. A derivation of the shape of a galaxy's rotation curve can be found through the formula:

Begeman's rotational data overlaid with a curve fit of the given equation. The fit yields a normalized sigma of 0.04 and a vmax of 152.9 km/s.
 vtan= vmax (ῶ0 r) / √(1+ ῶ02 r2).

which is the result of the metric in a curved space, namely:

 ds2 = c2 dt2 / (γω2)   - dr2 - γω22.


 γω = √(1+ ῶ02 r2).

Another consequence of this discovery is that both the flat rotation curve and spiral shape of NGC3198 is explained without having to resort to using dark matter or MOND.


  1. 1.0 1.1 Binney, J (2008). Galactic Dynamics. Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-13026-2. Search this book on Amazon.com Logo.png
  2. 2.0 2.1 Mathewson, D S (1992). "A southern sky survey of the peculiar velocities of 1355 spiral galaxies". apj. 81: 413.
  3. Abell, G (1975). Exploration of the Universe. New York: Holt, Rinehart and Winston. p. 621. ISBN 0030759501. Search this book on Amazon.com Logo.png
  4. Galaxies
  5. Rout, Bruce; Rout, Cameron (1 June 2016). "An Analytic Mathematical Model to Explain the Spiral Structure and Rotation Curve of NGC 3198". 228: 103.07.
  6. Shapley, Harlow; Curtis, Heber D., "79. The Scale of the Universe", A Source Book in Astronomy and Astrophysics, 1900–1975, Harvard University Press, ISBN 9780674366688, retrieved 2019-03-11
  7. leavitt, Henrietta (1912). "Periods of 25 Variable Stars in the Small Magellanic Cloud" (PDF). Harvard College Observatory Circular. 173: 1–3.
  8. Hubble, E. P. (1926-12-01). "Extragalactic nebulae". The Astrophysical Journal. 64. doi:10.1086/143018. ISSN 0004-637X.
  9. Ferrarese,, Laura; Ford, Holland C.; Huchra, John; Kennicutt, Jr., Robert C.; Mould, Jeremy R.; Sakai, Shoko; Freedman, Wendy L.; Stetson, Peter B.; Madore, Barry F. (2000). "A Database of Cepheid Distance Moduli and Tip of the Red Giant Branch, Globular Cluster Luminosity Function, Planetary Nebula Luminosity Function, and Surface Brightness Fluctuation Data Useful for Distance Determinations". The Astrophysical Journal Supplement Series. 128 (2): 431–459. doi:10.1086/313391. ISSN 0067-0049.
  10. Tulley, R.G. (1977). "A New Method of Determining Distances to Galaxies". Astronomy and Astrophysics. 54 (3): 661–673.
  11. three-line algebraic calculation
  12. Weisstein, Eric W. "Archimedes' Spiral". MathWorld.
  13. O'Meara, Stephen. The secret deep. Cambridge University Press. pp. 186–188. ISBN 978-0521198769. Search this book on Amazon.com Logo.png
  14. Zaveri, Vikram H. (15 May 2008). "Periodic relativity: deflection of light, acceleration, rotation curves".
  15. Ratcliffe, Hilton (February 1, 2010). The static universe : exploding the myth of cosmic expansion. Apeiron. p. 3. ISBN 9780986492624. Search this book on Amazon.com Logo.png

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