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Cepheid Variables

Planetary Nebulae Luminosity Function

Tully-Fisher Relation

VLBI

TypeIa Supernovae

Sunyaev-Zeldovich Effect

Gravitationally Lensed Quasars

A standard candle is a class of astrophysical objects, such as supernovae or variable stars, which have known luminosity due to some characteristic quality possessed by the entire class of objects. Thus, if an extremely distant object can be identified as a standard candle then the absolute magnitude M (luminosity) of that object is known. Knowing the absolute magnitude, the distance D (in cm) can be calculated from the apparent magnitude m as shown in the formula below.

m = M - 97.5 + 5xlog(D) ---------- (1)

Figure 01 below shows the different kind of "standard Candles", including the Cepheid Variables (star), the planetary nebulae, and the Type Ia supernovae (together with other distance measuring tools).

- Parallax - The parallax of a star is the angle subtended at the star by the radius of the Earth's orbit (Figure 02). Distance to the star can be computed by the formula:

d = 206265/p

where the distance d is in unit of a.u., and the parallax p in second of arc.

The usefulness of this method is limited by the resolution of the apparent positions to a distance of about 100 light years.

A parsec is the distance corresponds to a parallax of 1 sec. of arc, i.e.,

1 parsec = 206265 a.u. = 3.26 light years. - Cepheid Variables - It was discovered in 1912 by the American astronomer Henrietta Leavitt that the period of the Cepheid
variables depends on the star's mean absolute magnitude; the more luminous the star, the longer the period (Figure 03a). From a graph or a formula it is possible to find the absolute magnitude of a Cepheid variable if its period is known. The distance to the star D can then be computed from Eq.(1). The Cepheid variables have proved extremely important for astronomy because they can be used to calculate the distance of objects far beyond the Milky Way. The same kind of relation is also applicable to the RR Lyrae stars. They are all about fifty times more luminous than the Sun, and they vary by about a magnitude in periods of between 0.2 and 1.2 days. Figure 03b shows the Cepheid Variable star RS Pup (behind the dark central stripe) surrounded by dusty reflection nebula. It is some 10 times more massive than the Sun and pulsates over a period of about 40 days. #### Figure 03a Period-Luminosity Relation

_{}#### Figure 03b Cepheid Star RS Pup

[view large image]

- TRGB (Tip of the Red Giant Branch) - This technique of measuring cosmic distance was discovered in the 21st century, here's some introductory comments :
- When the main sequence star at the verge of using up the hydrogen fuel, its size expands in a phase called "Red Giant Branch" (RGB), which terminates at the point when helium burning commences. At this point, the size of the star reaches its maximum and ends with a flash (the helium flash).
- For low mass ( < 2 M
_{Sun}) main sequence star, the helium flash would occur at a particular absolute magnitude of M ~ - 4 corresponding to about 5x10^{3}L_{Sun}(see Figure 03c diagrams a, and b) and color index of (B - V) ~ +1.7, or (V - I) ~ +1.6 corresponding to a surface temperature of about 3000 K. - Such peculiar point of the star can be relatively easy to pick out from the globular clusters in the halo of galaxy. For example, the TRGB can be located in the HR diagram of M5 (Figure 3c, diagram c), a globular cluster in the halo of the Milky Way. Converting the apparent visual magnitude m
_{v}to m_{I}from the color indices, we obtain the distance D = 24 kly by applying Eq.(1). This value is very close to the 24.5 kly quoted by most literatures. For more distant galaxy, the tipping point can be identified by using the I-band, i.e., looking for (V - I) ~ +1.6 (the very red object). - The TRGB methodology has been applied to many distant galaxies as shown in Figure 3c, diagram d. it complements the measurement by using the Cepheid Variables. The latest data produce a Hubble constant H
_{0}~ 70 (km/sec)/Mpc which sits right at the middle between the contentious measurements from nearby astronomical objects and

#### Figure 03c TRGB

_{}the faraway CMB (see "Hubble Constant, 2018", "New Measure of Hubble Constant", and click the little icon _{}).

## Figure 02 Parallax |

N(M) ~ e ^{0.307M}{1 - e^{3(M* - M)}} ---------- (2)where N is the number of planetary nebula with absolute magnitude M, and M* represents the cutoff magnitude (there are no PN brighter than this), which is about -4.48. Thus, if we have a sample of PN from a given galaxy satisfying the PNLF as shown in Eq.(2), then we can obtain | |

## Figure 04 PNLF |
the apparent magnitude m* corresponding to the cutoff magnitude M*, and the distance D can be computed from Eq.(1). Figure 04 shows the theoretical and observed PNLF for M31. |

where

magnitude M can be expressed in term of V, which can be measured from the line width of the 21-centimeter line emitted by the hydrogen cloud. This is known as the Tully-Fisher relation as shown in Eq.(4): M = 4.8 - 2.5xlog(L/L _{sun}) ~ 10xlog(V) ---------- (4)The distance D can be calculated by Eq.(1) once the apparent magnitude m of the galaxy is known. Figure 05 is an example of a Tully-Fisher relation between M and log(V). Each data point corresponds to a galaxy within the cluster of galaxies. The solid line represents the theoretical best fit. | |

## Figure 05 Tully-Fisher Relation [view large image] |

radio signal - swirling in a disk of gas at the galaxy¹s center. Combining the maser's intrinsic speed (inferred from Doppler shift of the maser emission)
with the apparent angular motion as seen from Earth yields the distance D via the formula: D = V / (d/dt) ---------- (5) where V is the intrinsic velocity of the H _{2}O molecular cloud measured from Doppler shift of the maser emission, and d/dt is the proper motion of the molecular cloud
| |

## Figure 06 M106 |
observed directly by the VLBI. Sometimes the proper motion of the jet is measured (instead of the molecular cloud) such as in the case of 3C273. |

same amount of energy, and studies of relatively nearby type Ia supernovae have shown that they reach almost the same peak brightness in every case. Therefore it can be used as "standard candles" to determine their true distance. Figure 07 is a Type Ia supernova observed in 1994. It is the bright spot on the lower left at the fringe of the galaxy. Figure 08 shows such binary system before the explosion. | ||

## Figure 07 Supernova Type Ia |
## Figure 08 Binary System [view large image] |
The absolute magnitude for the Type Ia supernovae has been calibrated to be M = -19.33 0.25 |

through the bremsstrahlung process. The x-ray luminosity is proportional to ^{2}S, where is the density of the gas, and S is the size of the cluster. While the Compton scattering process transfer energy between the gas and the photons of the cosmic microwave background, effectively casting a shadow on the CMBR. The cluster is dimmer when observed at low frequencies but brighter at high frequencies, relative to the CMBR as shown by the dotted line in Figure 09. The magnitude of this effect is proportional to S. The x-ray luminosity and the microwave variation are combined to obtain a direct measurement of the size S of the region containing the hot gas. While direct observation gives the angular size . Thus the distance D can be computed by the formula:D = S / ---------- (6) | |

## Figure 09 SZ Effect |

- The time delay -- which means the intensity of the background source must vary sharply, by large amounts.
- The angular separation of the images of the background source.
- the relative distances of lens and background source (we can use the redshift for this).
- the mass of the lens, and its distribution (a point mass, or uniform sphere, or a sphere with varying density, or an ellipsoid).
| ||

## Figure 10 Lensed Quasar |
## Figure 11 Time Delay |