# Astronomy calculator: light time, magnitudes and redshift

> Light travel time for any distance, apparent and absolute magnitude from the distance modulus, and redshift to recession velocity (relativistic Doppler).

Versão interativa: https://www.calcopenly.com/pt/science/astronomy-calculator
Tema: Calculadoras científicas

Light travel time is the distance divided by the exact speed of light, 299,792,458 m/s. The distance modulus, m − M = 5 log₁₀(d/10 pc) + A, links a star's apparent magnitude m, its absolute magnitude M (how bright it would look from 10 parsecs), its distance d and the dimming A by dust; give any two of m, M and d to get the third. Redshift z converts to velocity with the relativistic Doppler formula 1 + z = √((1 + β)/(1 − β)), where β = v/c.

The default shows sunlight taking 8 min 19 s to cross 1 au (149,597,870.7 km), so we see the Sun as it was that long ago. Sirius, at apparent magnitude −1.46 and 2.64 parsecs, has absolute magnitude 1.43; from the same distance the Sun would shine at magnitude 1.94.

The Doppler formula treats redshift as motion through space. Beyond z ≈ 0.1 a galaxy's redshift comes mainly from cosmic expansion, so the velocity shown is not its recession speed and its distance needs a cosmological model.

## Dados

- **Calculate** (opções: Light travel time, Magnitudes, Redshift)
- **Distance to** (opções: Moon, Sun, Proxima Centauri, Galactic centre, Andromeda galaxy, Enter a distance)
- **Distance**
- **Solve for** (opções: Apparent m, Absolute M, Distance)
- **Apparent magnitude m**: Sirius, the brightest star in the night sky, is −1.46
- **Absolute magnitude M**: The Sun's absolute visual magnitude is 4.83
- **Distance**: Sirius is 2.64 pc away; absolute magnitude is defined at 10 pc
- **Extinction A (dimming by dust)**
- **Given** (opções: Redshift z, Velocity)
- **Redshift z**
- **Recession velocity**

## Resultados

- Light travel time — resultado principal
- Light travel time (years)
- Light travel time (min)
- Distance (km)
- Distance (ly)
- Apparent magnitude m
- Absolute magnitude M
- Distance (pc)
- Distance modulus m − M
- Brightness relative to a star at 10 pc
- Redshift z
- Recession velocity (relativistic Doppler) (km/s)
- Velocity as a fraction of c
- Low-speed approximation cz (km/s)

## Fórmula

$$
t = \frac{d}{c};\qquad m - M = 5\log_{10}\frac{d}{10\ \mathrm{pc}} + A;\qquad 1 + z = \sqrt{\frac{1 + \beta}{1 - \beta}},\ \beta = \frac{v}{c}
$$

## Exemplos resolvidos

### Sunlight takes about 8 min 19 s

- Calculate: Light travel time
- Distance to: Sun
- **Light travel time: 8.31675 min**
- **Distance: 149,598,000 km**
- Fonte de verificação: Python 3.8 decimal: 149597870700 m / 299792458 m/s = 499.00478 s (IAU au, exact c)

### Moon: 1.28 light-seconds

- Calculate: Light travel time
- Distance to: Moon
- **Light travel time: 1.28 s**
- Fonte de verificação: Python 3.8 decimal: 384400 km / c = 1.2822204 s

### One light-year takes one Julian year

- Calculate: Light travel time
- Distance to: Enter a distance
- Distance: 1 ly
- **Light travel time: 1 year**
- Fonte de verificação: IAU definition: 1 ly = c × 365.25 d

### Apparent magnitude of the Sun

- Calculate: Magnitudes
- Solve for: Apparent m
- Absolute magnitude M: 4.83
- Distance: 1 au
- **Apparent magnitude m: -26.742**
- **Distance modulus m − M: -31.572**
- Fonte de verificação: Python 3.8 decimal: 4.83 + 5 log₁₀(4.848137e-6/10) = −26.742 (observed −26.74)

### Absolute magnitude of Sirius (m −1.46 at 2.64 pc)

- Calculate: Magnitudes
- Solve for: Absolute M
- Apparent magnitude m: -1.46
- Distance: 2.64 pc
- **Absolute magnitude M: 1.432**
- Fonte de verificação: Python 3.8 decimal: −1.46 − 5 log₁₀(0.264) = 1.432 (published M ≈ 1.42)

### Distance modulus 10 means 1 kpc

- Calculate: Magnitudes
- Solve for: Distance
- Apparent magnitude m: 10
- Absolute magnitude M: 0
- **Distance: 1,000 pc**
- Fonte de verificação: d = 10^(1 + (m − M)/5) pc = 10³

## Perguntas

### How long does light take to reach Earth from the Sun?

8 minutes 19 seconds on average (499.0 s): 1 astronomical unit is exactly 149,597,870,700 m (IAU 2012) and light travels exactly 299,792,458 m/s. Earth's distance from the Sun varies from about 147.1 to 152.1 million km, so the delay ranges from roughly 8 min 11 s in January to 8 min 27 s in July. Moonlight takes 1.28 s.

### What is the difference between apparent and absolute magnitude?

Apparent magnitude m is how bright a star looks from Earth; absolute magnitude M is how bright it would look from a standard 10 parsecs (32.6 light-years). The scale runs backwards and is logarithmic: 5 magnitudes is exactly a factor of 100 in brightness, so 1 magnitude is 100^(1/5) ≈ 2.512 times. The Sun is m = −26.74 from Earth but only M = 4.83.

### How do you calculate distance from magnitudes?

d = 10^((m − M − A + 5)/5) parsecs, the distance modulus solved for d. A star with m − M = 10 and no dust lies at 10³ = 1,000 pc, about 3,262 light-years. Every 5 magnitudes of difference multiplies the distance by 10; Cepheid variables and Type Ia supernovae, whose absolute magnitudes are known, are measured this way.

### How do you convert redshift to velocity?

Use the relativistic Doppler formula, β = ((1 + z)² − 1)/((1 + z)² + 1), then v = βc. At z = 0.1 that gives 28,487 km/s, 5% below the simple estimate cz = 29,979 km/s, and at z = 1 it gives 0.6c, not c. For distant galaxies the redshift is cosmological, so a Hubble-law or ΛCDM calculation replaces this formula.

### How far is a light-year?

9,460,730,472,580.8 km, the distance light travels in one Julian year of 365.25 days, as defined by the IAU. A parsec is 3.2616 light-years, or 648,000/π astronomical units (IAU 2015 Resolution B2). Proxima Centauri, the nearest star after the Sun, is 4.2465 light-years away, so its light is 4.25 years old when it arrives.

### Qual é a precisão de “Astronomy calculator: light time, magnitudes and redshift”?

A precisão depende dos dados inseridos e das hipóteses do método. O cálculo decimal usa 50 algarismos significativos, mas estimativas, métodos numéricos e dados de origem podem ter menor precisão; o arredondamento exibido não elimina essas limitações. Exemplos resolvidos verificados com fontes independentes: 9. Por exemplo, “Sunlight takes about 8 min 19 s” é verificado com Python 3.8 decimal: 149597870700 m / 299792458 m/s = 499.00478 s (IAU au, exact c).

### De onde vem o método?

IAU 2012 B2 (astronomical unit), IAU 2015 B2 (parsec = 648000/π au); c exact (SI); OpenStax Astronomy 2e, §17.1 The brightness of stars (magnitudes) and §19.1 Fundamental measures of distance; OpenStax University Physics Volume 3, §5.7 Doppler effect for light.

## Fontes

- [IAU 2012 B2 (astronomical unit), IAU 2015 B2 (parsec = 648000/π au); c exact (SI)](https://www.iau.org/static/resolutions/IAU2012_English.pdf)
- [OpenStax Astronomy 2e, §17.1 The brightness of stars (magnitudes) and §19.1 Fundamental measures of distance](https://openstax.org/books/astronomy-2e/pages/17-1-the-brightness-of-stars)
- [OpenStax University Physics Volume 3, §5.7 Doppler effect for light](https://openstax.org/books/university-physics-volume-3/pages/5-7-doppler-effect-for-light)