# 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).

Phiên bản tương tác: https://www.calcopenly.com/vi/science/astronomy-calculator
Chủ đề: Máy tính khoa học theo chủ đề

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.

## Dữ liệu đầu vào

- **Calculate** (lựa chọn: Light travel time, Magnitudes, Redshift)
- **Distance to** (lựa chọn: Moon, Sun, Proxima Centauri, Galactic centre, Andromeda galaxy, Enter a distance)
- **Distance**
- **Solve for** (lựa chọn: 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** (lựa chọn: Redshift z, Velocity)
- **Redshift z**
- **Recession velocity**

## Kết quả

- Light travel time — kết quả chính
- 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)

## Công thức

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

## Ví dụ có lời giải

### 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**
- Nguồn đối chiếu: 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**
- Nguồn đối chiếu: 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**
- Nguồn đối chiếu: 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**
- Nguồn đối chiếu: 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**
- Nguồn đối chiếu: 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**
- Nguồn đối chiếu: d = 10^(1 + (m − M)/5) pc = 10³

## Câu hỏi

### 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.

### “Astronomy calculator: light time, magnitudes and redshift” chính xác đến mức nào?

Độ chính xác phụ thuộc vào dữ liệu nhập và giả định của phương pháp. Phép tính thập phân dùng 50 chữ số có nghĩa, nhưng ước lượng, phương pháp số và dữ liệu nguồn có thể kém chính xác hơn; làm tròn khi hiển thị không loại bỏ các giới hạn đó. Ví dụ có lời giải đã đối chiếu với nguồn độc lập: 9. Ví dụ, “Sunlight takes about 8 min 19 s” được kiểm tra bằng Python 3.8 decimal: 149597870700 m / 299792458 m/s = 499.00478 s (IAU au, exact c).

### Phương pháp này lấy từ đâu?

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.

## Nguồn

- [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)