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9.7 Option – Astrophysics: 4. Determining distance
Extract from Physics Stage 6 Syllabus (Amended
October 2002). © Board of Studies, NSW.
[Edit: 16 Oct 04]
Prior learning:
Preliminary module 8.2 The World Communicates
Preliminary module 8.5 The Cosmic Engine
define absolute and apparent magnitude
- Apparent magnitude (m) is a relative
measure of how bright a star appears to an observer on
Earth. Magnitude is expressed as a number which increases
as apparent brightness decreases. The brightest visible
stars known to the ancients were arbitrarily given
magnitude 1, while the faintest visible stars were given
magnitude 6. The magnitude scale is now arranged such that
an increase in magnitude by 1.0 represents a decrease in
apparent brightness by a factor of 2.512, and a star of
magnitude m is 100 times fainter than a star of
magnitude m-5. The brightest visible
stars, other than the Sun, have apparent magnitude between
0 and –1, while telescopic stars have apparent
magnitude greater than 6. Apparent magnitude is sometimes
called apparent brightness.
- Absolute magnitude (M) is a relative measure of the amount of light a star radiates into space, that is, a measure of the star’s true luminosity. Absolute magnitude uses the same scale as apparent magnitude, and is arbitrarily defined to be the same as the apparent magnitude the star would have to an observer at a distance of 10 parsecs from the star. The largest negative numbers correspond to the most luminous stars. Absolute magnitude is sometimes called absolute brightness.
VIRTUAL EXPERIMENT 7 Calculating Absolute Magnitude
for Sirius A 
Brian von Konsky, Curtin University
of Technology, WA. A virtual experiment on calculating the
absolute magnitude of Sirius A. (This web site was last
checked on 15 August 2006.)
solve
problems and analyse
information using 
and 
to calculate
the absolute or apparent magnitude of stars using data and a
reference star
- To solve problems using these
equations you will need to rearranging equations involving
exponentials and logarithms. If you have problems doing
this follow the worked samples below to see which strategy
has been selected. If you still have difficulty, see your
teacher.
- You may be required to analyse
information presented in tables of astronomical
data including the known apparent and absolute magnitudes
for various stars. Make certain you understand which data
is required by a problem, and accurately extract that
information from the table.
- When applying the equations to calculate magnitudes and magnitude ratios for stars, remember that distance (d) is always measured in parsecs, and remember to use log to the base 10 (log10).
Sample problem 1
As seen from earth, the intensity of one star is three times that of another. Find the difference in their magnitudes to 2 significant figures.
Solution:
Since the ratio of their intensities is 3:1, then 
So you need to solve 
Taking logarithms to base 10 of both sides, we get 
That is, the magnitude difference 
= 
to 2 significant figures.
Sample problem 2
A star has an apparent magnitude of 5 but an absolute magnitude of -2. How far away is it?
Solution:
Using with 
and 
, we get 
or 
.
This gives the distance to the star as 
.
explain how the concept of magnitude can be used to determine the distance to a celestial object
- The concept of magnitude is used to indicate the
brightness of a star. Apparent magnitude (m)
describes how bright the star appears to an observer on
Earth. This number varies with the distance of the star
from the observer: the further the observer is from a star,
the fainter the star appears and the greater its magnitude
number. This is because the brightness of a star varies
inversely with the square of the distance from the star
(the inverse square law). Apparent magnitude is directly
measured from Earth.
- Absolute magnitude (M) describes the brightness the
star would have to an observer at a fixed distance of 10
parsecs from the star. This number is fixed because it is
taken at a fixed distance from the star. Absolute magnitude
is estimated for distant stars by comparison with reference
stars of the same spectral class and of known distance. If
a star is further away than 10 pc, its apparent magnitude
m is larger than its absolute magnitude M, because
the star appears fainter at the greater distance. If closer
than 10 pc, it would appear brighter and m would
be smaller than M.
- The amount by which m and M differ depends on the distance to the star. If both m and M are known, their difference (the distance modulus m – M) can be used to calculate the distance to the star using the distance modulus equation:
outline spectroscopic parallax
- Spectroscopic parallax is a technique for calculating
the distance to a star by comparing the spectroscopically
determined absolute magnitude with the apparent magnitude.
The term “parallax” is used only as an analogy
for distance.
- The broad shape of a star’s spectrum gives us its
temperature or colour class, which can be located on the
horizontal axis of a Hertzsprung-Russell
diagram. The fine details of the lines present
in the spectrum tell us its luminosity class. By drawing a
vertical line up from the position on the horizontal
spectral class axis until it intercepts with the luminosity
class, we can read off the star’s luminosity on the
vertical axis. If we know the luminosity of a star we can
calculate its absolute magnitude.
- The absolute magnitude can then be compared with the
directly-measured apparent magnitude to find the distance
to the star using . This technique is accurate to a
distance of about 10 megaparsecs (10 Mpc).
perform an investigation to demonstrate the use of filters for photometric measurements
- Your teacher may have planned an investigation for you,
or you could follow a procedure such as that outlined
below. When performing this investigation,
choose a light source that will allow you to demonstrate
the effect of a range of different coloured filters. Take
care to make measurements accurately and record your
findings systematically. Analyse the effect of changing the
colour of the filter in relation to the spectra of stars
and to the types of filters used by astronomers.
- To demonstrate the use of filters, show that there is a quantitative difference in the light from the same source transmitted through different filters.
Sample procedure
Produce simulated starlight from the incandescent lamp in a ray box kit, commonly available in school science laboratories. This has the advantage that coloured filters mounted in 35 mm slide frames can easily be inserted in the light path. If this is not available, filters can be held by hand in front of any incandescent lamp.
Use a light intensity probe attached to a datalogger to measure the intensity of light at a set distance from the lamp. Set the datalogger to operate in manual or “snapshot” mode. A photographer’s hand-held light meter is a suitable alternative to measure light intensity.
Place different coloured filters, one at a time, between the lamp and the light probe. For each filter, measure the intensity of light with the datalogger. You should note that the filters used in photometry, unlike those in a ray box kit, transmit a carefully calibrated range of frequencies.
For each filter, also observe the light through a hand-held spectroscope to see qualitatively what effect the filter has on the spectrum of white light produced by the lamp. Use the in-built scale to measure the range of wavelengths transmitted.
Record all your observations systematically in a suitable table. Compare your qualitative and quantitative observations for different filters.
Use your observations to predict the effect of different filters on the measurement of apparent magnitude of stars of different spectral type.
Photometry (Astronomy) ., Wikipedia.
explain how two-colour values (ie colour index, B-V) are obtained and why they are useful
- The photographic emulsions used in astronomy are most
sensitive to blue-violet light, whilst the human eye is
most sensitive to yellow-green light and less sensitive to
blue light. This means that, for example, a hot blue star
would appear brighter (lower magnitude) on a photographic
plate than to the unaided human eye because its maximum
intensity is in the blue region of the spectrum.
- Blue or photographic magnitude (B) of
a star is the magnitude measured through a blue filter
which allows only those wavelengths centred in a range
around 440 nm to pass. The visual
magnitude (V) of a star is the magnitude measured
through a yellow-green filter which allows only wavelengths
around 550 nm to pass. Alternatively, two types of colour
sensitive photographic film emulsion are used – a
blue-sensitive film that provides the photographic
magnitude (B) and a yellow-sensitive film that
provides the photovisual magnitude (V), which
replaces the old visual magnitude.
- Measuring apparent magnitude in two colours is useful
as a quick way to find the spectral class of a star without
using a spectroscope. Colour index (C.I.)
of a star is defined as the difference between its
photographic magnitude and its visual magnitude: C.I. = B
– V. Colour index ranges from –0.6 for hot blue
spectral class O stars to +2.0 for cool red spectral class
M stars, and is defined to be zero for white spectral class
A stars.
- Today the range of filters used in astronomical photometry has been extended to include ultraviolet, blue, yellow-green, red and infrared filters.
identify data sources, gather, process and present information to assess the impact of improvements in measurement technologies on our understanding of celestial objects
- To identify data sources that might be
useful, think about what information you are trying to
obtain and present. This may give you search terms to enter
into an Internet search engine or words to use when
scanning written material. Limit your search specifically
to technologies that allow measurement of some quantity,
such as distance, frequency or wavelength of features of
stellar spectra, surface temperature, brightness, etc. Do
not focus only on new technologies at the expense of
improvements in traditional means of astronomical
measurement.
- Try to gather information from a range
of resources, including popular scientific journals,
digital technologies like CD-ROMS and the Internet. Collate
information on at least TWO different technologies from
several different sources and produce a summary for each.
Be careful not to plagiarise any source, but to synthesise
the information and express it in your own words.
- Remember that information on the Internet is not
subject to peer review so that people’s
interpretation of new observations can be speculative.
(This happens even when work is peer reviewed!).
Process the information you find for each
technology to ensure its reliability by comparing it with
authoritative sources such as university-based or
government web sites.
- Re-read the dot point carefully to understand exactly
what it asks you to do: “assess the
impact of ...”. Structure your answer carefully. For
each chosen technology you should identify and describe the
technology or technological improvement, describe how the
technology contributes to our understanding of celestial
objects, and make a judgement of the extent to which this
technological improvement has increased our understanding.
- Select a suitable medium, format and text type(s) in which to present your work. Consider the appropriate use of pictures, diagrams or graphs. As your work will have been collected from a wide range of sources, pay particular attention to the way you acknowledge those sources.
Sample topics
One obvious new technology involving measurement is the use of electronic data collection and digital storage. Charge-coupled devices (CCDs) and computerised technology have enabled incredible leaps in the quantity and quality of data collected.
Some other things to search for on the Internet that would admirably demonstrate the impact of new technology on our understanding of celestial objects are:
- the Cosmic Background Explorer
- the Wilkinson Microwave Anisotropy Probe
- the Hubble Space Telescope
- the Chandra X-ray Telescope
- and any of the NASA planetary probes
History of Astronomy: Topics:
Instruments Dr Wolfgang R. Dick, Potsdam,
Germany.
Research Interests and History Dr Michael
Stanley Bessell, ANU Canberra and Siding Springs and Mt
Stromlo Observatories. (These web sites were last checked
on 15 August 2006)
describe the advantages of photoelectric technologies over photographic methods for photometry
- Photographic photometry utilises visual
comparisons between the images of stars on photographic
film. The diameter of each star’s image on the film
is related to its magnitude. It is possible to obtain
photometry for thousands of stars from a single photograph
using this technique. Lasers can be used to scan the plate
to produce a digitised image which can then be analysed.
- Photoelectric photometry uses a
photomultiplier to convert weak light into a measurable
electric current. Light from a single star falls through a
pinhole onto a photocathode, causing electrons to be
ejected in proportion to the intensity of the light. A
photomultiplier produces a pulse of current for every
electron ejected, and pulses are counted to produce an
digital signal which can analysed by a computer. Several
photomultipliers can be used simultaneously to measure the
light from different stars.
-
There are many advantages of photoelectric technologies
over the older photographic techniques:
- CCDs have a more uniform response across the visible spectrum than photographic film does, and corrections must be made for this in photographic photometry.
- CCDs are sensitive to a wider range of wavelengths than photographic film, particularly in the infrared.
- CCDs and photomultipliers are more sensitive than photographic film.
- Modern CCD arrays have a greater resolution than photographic film.
- Information can be collected much more quickly with electronic sensors than from photographs.
- Information can be collected remotely and transmitted digitally.
- Data can be processed more easily and quickly.
- There is more scope for a greater level of analysis because of the increased quantity of data.
- Photoelectric photometry allows for a faster and more accurate measurement of magnitude than photographic photometry.
Filters and CCDs , Anglo-Australian Observatory.
(This web site was last checked on 15 August 2006)
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