<p>There's no significant difference over a distance of a hundred light years or so.</p>
<p>The gravitational time dilation can (ignoring rotational terms) be calculated as the velocity time dilation of the escape velocity at that location. In the neighbourhood of the sun, the galactic escape velocity is about 550.9 km/s.
<a href="https://i.sstatic.net/mCbRr.png" rel="noreferrer"><img src="https://i.sstatic.net/mCbRr.png" alt="enter image description here" /></a></p>
<p><a href="https://arxiv.org/pdf/1408.1787.pdf" rel="noreferrer">https://arxiv.org/pdf/1408.1787.pdf</a></p>
<p>Eyeballing the graph suggests that for every kiloparsec, the escape velocity would increase by 10km/s, So 100 lightyears = 0.03 kpc closer to the galactic centre would have an escape velocity of 551.2 km/s</p>
<p>These would mean that at the orbit of the sun, the gravitational time dilation is 1.000001688, and 100 light-years closer to the galactic centre it is 1.000001690,  a difference of no practical importance for a space-faring.</p>

CC BY-SA 4.0
2023-03-15T23:07:19.810Z
2023-03-15T23:07:19.810Z
James K
<p>As others point out, the differential effects of the different positions in the Galactic potential are of order 1 part in a billion.</p>
<p>A bigger effect is the gravitational potential of the star. If the question is how much slower does time pass at the (surface of) a star compared with the Earth, then the gravitational redshift is approximately <span class="math-container">$0.5(M/R)$</span> km/s, where <span class="math-container">$M$</span> and <span class="math-container">$R$</span> are the mass and radius in solar units.</p>
<p>This amounts to a difference in clock speed of <span class="math-container">$500/3\times 10^8$</span>, where the denominator is the speed of light in m/s. i.e. Clocks run faster on Earth by 1 part in about 600,000.</p>
<p>The time dilation due to relative speeds is a smaller effect. If the standard deviation of nearby stars with respect to the Sun is 30 km/s, then the RMS time dilation is about 1 part in 200 million.</p>

CC BY-SA 4.0
2023-03-15T23:52:42.700Z
2023-03-16T00:03:57.817Z
2023-03-16T00:03:57.817Z
ProfRob
<h4>Nearby stars</h4>
<p>For such a small distance of <span class="math-container">$\sim 100$</span> light years, you can approximate that the Galaxy's gravitational field <span class="math-container">$g$</span> is uniform. On average, stars near the Sun orbit at speeds <span class="math-container">$v\simeq 220\text{ km/s}$</span> at a radius of <span class="math-container">$r\simeq8.2\text{ kpc}$</span>. Assuming circular orbits, that corresponds to a gravitational acceleration of <span class="math-container">$g=v^2/r=2.0\times 10^{-11}\ c^2/\text{lyr}$</span>; the unit here is squared speed of light per light year, which is appropriate for this problem.</p>
<p>For an observer closer to the Galactic center by distance <span class="math-container">$\Delta r$</span>, time &quot;passes slower&quot; by the fraction <span class="math-container">$g\Delta r/c^2=2.0\times 10^{-11}(\Delta r/\text{lyr})$</span> in the weak-field limit. If <span class="math-container">$\Delta r=100~\text{lyr}$</span>, time passes slower by <span class="math-container">$2.0\times 10^{-9}$</span>, or a couple of parts per billion (a couple of nanoseconds per second).</p>
<h4>Distant stars</h4>
<p>More generally, the fractional time dilation between two observers is just the gravitational potential difference between them, <span class="math-container">$\Delta\Phi/c^2$</span>. Here's a spherical approximation of the Milky Way's gravitational potential, calculated using the spherical mass model of <a href="https://arxiv.org/abs/1911.04557" rel="noreferrer">Cautun et al. (2020)</a>.
<a href="https://i.sstatic.net/Z1ax8.png" rel="noreferrer"><img src="https://i.sstatic.net/Z1ax8.png" alt="galactic potential" /></a>
The unit here is <span class="math-container">$10^{-6}c^2$</span>, so if the vertical axis differs by <span class="math-container">$1$</span> between two radius values, that means the gravitational time dilation between observers at those radii is one part in a million (one microsecond per second).</p>

CC BY-SA 4.0
2023-03-16T18:50:57.350Z
"a difference of 1 between two radii" or "a difference of 1 order of magnitude between 2 radii"?
Caridorc
CC BY-SA 4.0
2023-03-16T18:52:41.383Z
@Caridorc A difference of $1$ on the vertical axis (not the horizontal axis)
Sten
CC BY-SA 4.0
2023-03-15T23:19:25.890Z
2023-03-16T19:06:17.113Z
2023-03-16T19:06:17.113Z
Sten
<p>I can use data from <a href="https://www.cosmos.esa.int/web/gaia/dr3" rel="noreferrer">Gaia Mission</a> to estimate radial velocity(from RVS) and proper motion(from parallax), to take into account special relativity. The remaining question is what data should I use to estimate gravitational effects on time flow.</p>
<p>For example, how much faster/slower time passes for a star that is hundred light years closer to the galactic center than Earth?</p>
<p>It is reasonable to assume that there are at least a few big components of the gravitational field there.</p>
<ul>
<li>The gravity of the star itself</li>
<li>The gravity of the surrounding galaxy.</li>
</ul>
<p>The star's gravity is more or less possible to estimate, by estimating its mass. Good sources of data for this are very welcome.</p>
<p>For the second component, I have no idea.</p>
<p>This is not a research question, so the result should be believable, but not necessarily scientifically correct.</p>
<p>The baseline is to use Neuton's gravity law for a massive disk with an average density close to the average density of the Milky Way. I suspect that there are some ways to improve this estimation.</p>
<hr />
<p>P.S. If there are better data sources for true velocities of the distant stars feel free to point them out.</p>

CC BY-SA 4.0
2023-03-16T15:32:11.927Z
See [Why isn't the center of the galaxy "younger" than the outer parts?](https://physics.stackexchange.com/questions/161453/why-isnt-the-center-of-the-galaxy-younger-than-the-outer-parts) Spoiler: it is younger, but not by much :-)
John Rennie
CC BY-SA 4.0
2023-03-16T17:12:33.810Z
@JohnRennie, thank you. This is a good example of what I am looking for
FrogOfJuly
<time-dilation><estimate>
How do I estimate the difference in the speed of time flow between Earth and a distant star?
FrogOfJuly
# How do I estimate the difference in the speed of time flow between Earth and a distant star?

(Asked by: username_0 on 2023-03-15)

I can use data from [Gaia Mission](https://www.cosmos.esa.int/web/gaia/dr3) to estimate radial velocity(from RVS) and proper motion(from parallax), to take into account special relativity. The remaining question is what data should I use to estimate gravitational effects on time flow.

For example, how much faster/slower time passes for a star that is hundred light years closer to the galactic center than Earth?

It is reasonable to assume that there are at least a few big components of the gravitational field there.

* The gravity of the star itself
* The gravity of the surrounding galaxy.

The star's gravity is more or less possible to estimate, by estimating its mass. Good sources of data for this are very welcome.

For the second component, I have no idea.

This is not a research question, so the result should be believable, but not necessarily scientifically correct.

The baseline is to use Neuton's gravity law for a massive disk with an average density close to the average density of the Milky Way. I suspect that there are some ways to improve this estimation.

---

P.S. If there are better data sources for true velocities of the distant stars feel free to point them out.

**Tags:** time-dilation, estimate
> **Comments:**
> **username_1**: See [Why isn't the center of the galaxy "younger" than the outer parts?](https://physics.stackexchange.com/questions/161453/why-isnt-the-center-of-the-galaxy-younger-than-the-outer-parts) Spoiler: it is younger, but not by much :-)  
> **username_0**: @JohnRennie, thank you. This is a good example of what I am looking for  

---

## 3 answers

### Answer by username_4 on 2023-03-15. Score: 12 | Accepted answer

#### Nearby stars

For such a small distance of $\sim 100$ light years, you can approximate that the Galaxy's gravitational field $g$ is uniform. On average, stars near the Sun orbit at speeds $v\simeq 220\text{ km/s}$ at a radius of $r\simeq8.2\text{ kpc}$. Assuming circular orbits, that corresponds to a gravitational acceleration of $g=v^2/r=2.0\times 10^{-11}\ c^2/\text{lyr}$; the unit here is squared speed of light per light year, which is appropriate for this problem.

For an observer closer to the Galactic center by distance $\Delta r$, time "passes slower" by the fraction $g\Delta r/c^2=2.0\times 10^{-11}(\Delta r/\text{lyr})$ in the weak-field limit. If $\Delta r=100~\text{lyr}$, time passes slower by $2.0\times 10^{-9}$, or a couple of parts per billion (a couple of nanoseconds per second).

#### Distant stars

More generally, the fractional time dilation between two observers is just the gravitational potential difference between them, $\Delta\Phi/c^2$. Here's a spherical approximation of the Milky Way's gravitational potential, calculated using the spherical mass model of [Cautun et al. (2020)](https://arxiv.org/abs/1911.04557).
[![galactic potential](https://i.sstatic.net/Z1ax8.png)](https://i.sstatic.net/Z1ax8.png)
The unit here is $10^{-6}c^2$, so if the vertical axis differs by $1$ between two radius values, that means the gravitational time dilation between observers at those radii is one part in a million (one microsecond per second).

> **Comments:**
> **username_5**: "a difference of 1 between two radii" or "a difference of 1 order of magnitude between 2 radii"?  
> **username_4**: @Caridorc A difference of $1$ on the vertical axis (not the horizontal axis)  

---

### Answer by username_2 on 2023-03-15. Score: 5

There's no significant difference over a distance of a hundred light years or so.

The gravitational time dilation can (ignoring rotational terms) be calculated as the velocity time dilation of the escape velocity at that location. In the neighbourhood of the sun, the galactic escape velocity is about 550.9 km/s.
[![enter image description here](https://i.sstatic.net/mCbRr.png)](https://i.sstatic.net/mCbRr.png)

<https://arxiv.org/pdf/1408.1787.pdf>

Eyeballing the graph suggests that for every kiloparsec, the escape velocity would increase by 10km/s, So 100 lightyears = 0.03 kpc closer to the galactic centre would have an escape velocity of 551.2 km/s

These would mean that at the orbit of the sun, the gravitational time dilation is 1.000001688, and 100 light-years closer to the galactic centre it is 1.000001690, a difference of no practical importance for a space-faring.


---

### Answer by username_3 on 2023-03-15. Score: 5

As others point out, the differential effects of the different positions in the Galactic potential are of order 1 part in a billion.

A bigger effect is the gravitational potential of the star. If the question is how much slower does time pass at the (surface of) a star compared with the Earth, then the gravitational redshift is approximately $0.5(M/R)$ km/s, where $M$ and $R$ are the mass and radius in solar units.

This amounts to a difference in clock speed of $500/3\times 10^8$, where the denominator is the speed of light in m/s. i.e. Clocks run faster on Earth by 1 part in about 600,000.

The time dilation due to relative speeds is a smaller effect. If the standard deviation of nearby stars with respect to the Sun is 30 km/s, then the RMS time dilation is about 1 part in 200 million.


---

