luminous distance

by Ted Frimet

how to ride a light beam

AAAP board member Gene Allen programmed outreach to a Scout Group, for Wednesday night, April 18th, at Bordentown, NJ. No problem, so I thought! These Scouts had an ulterior motive, though. The hidden secret agenda was star brightness and distance calculations. That is one tall order for any neophyte Amateur. Maybe not for our more experienced membership, of course! I realized that, for me, I had better break out the textbooks, and learn to turn a page. It turned out to be a lot of page turning!

I decided to brief the Scouts on a little history, with this essay, followed by some vocabulary. Then feature a basic, intuitive view into star brightness. And close with a luminosity distance calculation.

Astronomers continue to use a system that is based upon Hipparchus. (1) This Greek astronomer established a magnitude method over 2,000 years ago. In a nutshell, the larger the number, the dimmer the star. His simple system accounted for stars between 1st and 6th magnitude.

We keep Hipparchus’ system in mind, even today, and with a twist. In modern times, we speak of magnitude jumps. A 1st-magnitude jump is a brightness change of 2.5 times. A 2nd-magnitude jump is another brightness change using an extra 2.5 factor.

Using basic mathematics, we calculate that a 3rd-magnitude jump is:

( 2.512 x 2.512 ) = 6.310 times brighter than a 1st magnitude star.

Sometime in the 19th century, Astronomers further refined the brightness scale. We can now accommodate stars that are fainter than 6th magnitude. Under dark skies our naked eye limits reach 6th magnitude. The Hubble Space Telescope (HST) can see +30. Our club’s 14 inch diameter telescope can see stars about half-magnitude of HST.

Just a few weeks ago, I was at a dark sky site, at Jenny Jump State Park, trying to observe The Pinwheel Galaxy. This galaxy has a Messier name of M101. Messier was an Astronomer that noted all the fuzzy things in the night sky. One of our members can point out M101’s location to you, with a laser. However, at around magnitude 9, it cannot be seen with the eye. Even with a 10 inch telescope, it is diffuse and hard to spot. (2) Another source describes this Ursa Major resident at magnitude 7.9 (3). I could not view this face-on galaxy with my 12 inch Newtownian telescope, that dark night.

I might venture we could use low power binoculars to see the very diffuse Pinwheel. My friend and budding Astrophotographer, Captain James DiPietro, US Army NG, member UACNJ, managed to capture images with 90 second exposures. The human eye, being quite different from an electronically assisted astronomy (EAA), cannot build up photons, like Captain DiPietro’s camera. For us, EAA is the best way to view M101.

I did, however, manage to see her sister, M51, the Whirlpool Galaxy. It has since become my favorite, at 8.96 magnitude (4). You might ask, if M101 is brighter than M51, why could I not see it? Diffuse objects in the night sky are very hard for an amateur to see. M51 is more dense, with a size of 11’ x 7’ (arc minutes), 27.9 light years away.

M101 is 8.31 magnitude, 29’ x 27’ size, and 22.2 light years. An object may be closer, or brighter. That doesn’t mean we can view it more easily.

If you want to study a diffuse galaxy, you don’t want to use your highest power objectives. Use lower power to take in a greater field of view (FOV). Ask one of our observers to show you the Andromeda galaxy.

How bright a star is, is defined by absolute magnitude. This is how bright a star appears at a standard distance of 32.6 lightyears. Our sun, Sol, has an apparent magnitude of [-26.7]. Sols’ absolute magnitude is only 4.8 (5).

Astronomers calculate distances by the parallax method. That is, stars appear to move, and shift behind our closer nearby stars. By using math, really by using triangular geometry, you can figure out a distance to a star. According to the Astronomy Education Center at the University of Nebraska-Lincoln, the parallax method is good for stars out to 500 light-years.

We use the distance modulus to calculate even further stellar distances. You can estimate the stellar distance by subtracting absolute magnitude (M) and apparent magnitudes (m) (6) (7). However that is only a first step. We need to know both magnitudes, and apply logarithm math.

We know how to estimate apparent magnitude (m). We could do it, just like Hipparchus, and modify according to modern use. How then do we get absolute magnitude (M)? We measure it using sophisticated software. Or we can look up our numbers in Astronomy reference books.

A brightest stars table, listing 286 stars in all, can be found in the RASC Observers Handbook 2018, USA edition (8). There you will find m & M, as well as distances tabulated in light-years.

A more hands-on approach is to record the M, yourself. For this, we need to become astrophotographers, and make use of well designed telescopes, under dark skies.

One AAAP asset that we subscribe to is the Skynet Robotic Network. The University of North Carolina, Chapel Hill hosts the Skynet telescope system. Afterglow is their post-processing program. We use this software to sample absolute magnitude from astrophotography data. Take a picture of a star, using Skynet, then use Afterglow. Put your cursor on your star and Afterglow will record the absolute magnitude.

Researching the web, you can find many places to learn from. Some are easy to grasp, and some, like the below link, is just out of my learning curve:

http://astro.wku.edu/labs/m100/mags.html

Can we make this more simpler? Yes, we can!

Light intensity decreases as the distance squared.

Basically, the farther out the star is, the less light will reach your eyes.

If a star is 3 times farther out, then the light is 9 times less intense.

If a star is 2 times farther out then the light is 4 times less. Get it?

At the end of the essay you will find online references, and a bibliography for further study.

Now, here is your first distance modulus math calculation:

m – M = (11.13 – 15.56) = -4.43

-4.43 ==> d

d = 1.300 parsecs

parsecs to light years conversion:

Distance = 1.3 parsecs = 4.24 light-years

parsecs to light years (9)

How we derive “d” from the magnitude difference is complex. If you like, you can read below. And we will discuss in detail how we get from m-M to d.

Here is some data (10), taken from a table of nearest stars from The Observers Handbook:

Proxima Centauri

apparent magnitude (m) = 11.13

absolute magnitude (M) = 15.56

ly (light years) = 4.24

Sirius (A)

m = -1.43

M = 1.47

ly = 8.58

Let’s calculate how far Proxima Centauri is. You may recall, from your Scout research, that this red dwarf star, is Sol’s closest neighbor. It is mentioned in this months essay (11), “go fly a kite”, as the Breakthough Starshot Intiative’s destination. Read more, here:

https://princetonastronomy.wordpress.com/2018/04/02/go-fly-a-kite/

Need the math? Ok. Here we go. Hang on!

Proxima Centauri m = 11.13 and M = 15.56

m – M = -5 + 5 log10(d)

11.13 – 15.56 = -5 + 5 log10(d)

-4.43 = 5log10(d) – 5

Using a wiki reference to isolate “d” (retrieved April 3, 2018)

https://en.wikipedia.org/wiki/Distance_modulus

Here, I mean that d = 10↑(distance modulus / 5) + 1

where “↑” means raised to the “power of (distance modulus / 5)”

then add 1 to the answer.

d = 10↑(-4.43/5) +1

d = 10↑(-.886) + 1

d = 0.1300169578 + 1

d = 1.3 parsecs

Distance = 1.3 parsecs = 4.24 light-years

If we check with our RASC reference, our distance to Proximus Centauri is confirmed.

We have calculated a distance to a star, by using the available light.

Now it is time to take notice that Sirius, the dog star, is much brighter than Sol’s closest stellar neighbor. Yet doing the math, we conclude that Sirius is 2/3rds (67%) farther away from us than Proximus Centauri. You have now proved that just because something is brighter, doesn’t necessarily mean it is closer. We will continue to plan on

dealing with star brightness as a function of distance. To do so, we must include both magnitudes types (little m & big M) in our calculations.

By happenstance, I had an opportunity to attend a lecture, this Friday afternoon, April 6th at The University of the Sciences, Philadelphia, Pennsylvania. Today’s guest lecturer was Erica Ellingson, PhD. Dr Ellingson is from the Department of Astrophysical & Planetary Sciences, University of Colorado; Fellow of the Center for Astrophysics & Space Astronomy. I was met there by fellow AAAP member and Program Chair, Ira Polans.

Regretfully, Philadelphia traffic barred me from Dr Ellingson’s first lecture. I did manage to squeeze in a slice of pizza, and the second seminar topic: Dark Energy and Cosmology. Although the topic was well presented and lucid, I’d like to bring out an important side note, touched upon by our lecturer: the Type 1A Nova.

During the seminar, I was reminded that there is a rare chance of a nova in galaxy. And if we group hundreds of galaxies together, for study, we will see many of them. One in particular is of stellar importance to luminous distance measurement. It is the Type 1A supernovae.

The absolute magnitude of a Type 1A is ALWAYS the same luminosity. However, the apparent magnitude, which varies by distance, is not always the same. If you apply the distance modulus math, you can calculate the distance to the parent galaxy. That is, you can tell the distances to stars, galaxies, and the great spaces between them all.

I would venture to say, if you hang around long enough, a Type 1A will show you how to ride a light beam; right up to the edge of what is the visual horizon of our 14.7 billion year old Universe.

Notes, resources and bibliography: (all links retrieved April 3, 2018).

There is an online distance modulus calculator hosted by University of Nebraska-Lincoln. Use it to check your work.

http://astro.unl.edu/naap/distance/distance_modulus.html

With greater ease we can review a Cornell University online document. It has plenty of math, and sample data to use.

http://www.astro.cornell.edu/academics/courses/astro1101/lectures6StellarDistancesRev1.pdf

For more information, you can read here, at Swinburne Astronomy Online:

http://astronomy.swin.edu.au/cosmos/D/Distance+Modulus

A second math approach to calculate luminosity distance (12):

https://en.wikipedia.org/wiki/Luminosity_distance

M = absolute magnitude

m = apparent magnitude

Luminosity distance: DL (written as D)

M = m-5(log10 D – 1)

D = 10↑(m-M/5) + 1

distance = 10↑(11.13 – 15.56)/5) + 1

distance = 10↑(-4.43 / 5) + 1

distance = 10↑(-.886) + 1

distance = .130 + 1

distance = 1.30 parsecs

https://www.metric-conversions.org/length/lightyears-to-parsecs.htm (13)

pc = ly * 0.30660

ly = pc/.30660

ly = 1.30 / .30660

ly = 4.24 ly

  1. Harrington, P. S. (2003). Star watch: The amateur astronomers guide to finding, observing, and learning about over 125 celestial objects. Hoboken, NJ: Wiley. pps7-8
  2. Burnham, R., Dyer, A., Garfinkle, R., George, M., Kanipe, J., Levy, D. H., & O’Bryne, D. (2002). A guide to advanced skywatching: The backyard astronomers guide to starhopping and exploring the universe. San Francisco, CA: Fog City Press., p234
  3. Sparrow, G. (2015). The stargazers handbook: The definitive field guide to the night sky. London: Quercus., p32
  4. Madore, B. F., & Steer, I. (2018). The Royal Astronomical Society of Canada Observer’s Handbook (2018 ed., 110th year of publication – RASC) (J. S. Edgar, Ed.). Canada: Webcom. Observer’s Handbook 2018 USA Edition. Galaxies Brightest and Nearest, p333
  5. Burnham, R., et al, ibid, p162
  6. Burnham, R., et al, ibid, p163
  7. Bishop, Roy (2018), RASC, et al, Some astronomical and physical data, p31
  8. Karmo, Toomas, Corbally, Chris, & Gray, Richard (2018), RASC, et al,
    The brightest stars, pps 275-283
  9. Google web search April 3, 2018: “Parsecs to Light Years” conversion:
    https://www.google.com/search?client=safari&rls=en&q=parsecs+to+light+years&ie=UTF-8&oe=UTF-8
  10. Henry, Todd J. (2018), RASC, et al, The nearest stars, p289
  11. Frimet, T. R. (2018, April 02). Go fly a kite (S. Agarwal & P. Ganti, Eds.). Retrieved April 03, 2018, from https://princetonastronomy.wordpress.com/2018/04/02/go-fly-a-kite/
  12. Luminosity distance. (2018, February 28). Retrieved April 03, 2018, from
    https://en.wikipedia.org/wiki/Luminosity_distance
  13. https://www.metric-conversions.org/length/lightyears-to-parsecs.htm Retrieved April 3, 2018. Wight Hat Ltd. ©2003-2018. Their page last updated: Thr 22 Mar 2018
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